Changes since the initial port
The following section lists changes to this file in mathlib3 and mathlib4 that occured after the initial port. Most recent changes are shown first. Hovering over a commit will show all commits associated with the same mathlib3 commit.
Changes in mathlib3
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Changes in mathlib3port
mathlib3
mathlib3port
bump to nightly-2024-04-09-08
mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
Diff
@@ -854,19 +854,19 @@ theorem adjoin_one : F⟮⟯ = ⊥ :=
-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
-#print IntermediateField.adjoin_int /-
+#print IntermediateField.adjoin_intCast /-
@[simp]
-theorem adjoin_int (n : ℤ) : F⟮⟯ = ⊥ :=
- adjoin_simple_eq_bot_iff.mpr (coe_int_mem ⊥ n)
-#align intermediate_field.adjoin_int IntermediateField.adjoin_int
+theorem adjoin_intCast (n : ℤ) : F⟮⟯ = ⊥ :=
+ adjoin_simple_eq_bot_iff.mpr (intCast_mem ⊥ n)
+#align intermediate_field.adjoin_int IntermediateField.adjoin_intCast
-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
-#print IntermediateField.adjoin_nat /-
+#print IntermediateField.adjoin_natCast /-
@[simp]
-theorem adjoin_nat (n : ℕ) : F⟮⟯ = ⊥ :=
- adjoin_simple_eq_bot_iff.mpr (coe_nat_mem ⊥ n)
-#align intermediate_field.adjoin_nat IntermediateField.adjoin_nat
+theorem adjoin_natCast (n : ℕ) : F⟮⟯ = ⊥ :=
+ adjoin_simple_eq_bot_iff.mpr (natCast_mem ⊥ n)
+#align intermediate_field.adjoin_nat IntermediateField.adjoin_natCast
-/
section AdjoinRank
bump to nightly-2024-04-06-08
mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
Diff
@@ -6,7 +6,7 @@ Authors: Thomas Browning, Patrick Lutz
import FieldTheory.IntermediateField
import FieldTheory.Separable
import FieldTheory.SplittingField.IsSplittingField
-import RingTheory.TensorProduct
+import LinearAlgebra.TensorProduct.Tower
#align_import field_theory.adjoin from "leanprover-community/mathlib"@"6b31d1eebd64eab86d5bd9936bfaada6ca8b5842"
bump to nightly-2024-03-17-08
mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
Diff
@@ -640,7 +640,7 @@ variable {F} {α}
theorem adjoin_algebraic_toSubalgebra {S : Set E} (hS : ∀ x ∈ S, IsAlgebraic F x) :
(IntermediateField.adjoin F S).toSubalgebra = Algebra.adjoin F S :=
by
- simp only [isAlgebraic_iff_isIntegral] at hS
+ simp only [isAlgebraic_iff_isIntegral] at hS
have : Algebra.IsIntegral F (Algebra.adjoin F S) := by
rwa [← le_integralClosure_iff_isIntegral, Algebra.adjoin_le_iff]
have := isField_of_isIntegral_of_isField' this (Field.toIsField F)
@@ -781,7 +781,7 @@ theorem exists_finset_of_mem_iSup {ι : Type _} {f : ι → IntermediateField F
(hx : x ∈ ⨆ i, f i) : ∃ s : Finset ι, x ∈ ⨆ i ∈ s, f i :=
by
have := (adjoin_simple_is_compact_element x).exists_finset_of_le_iSup (IntermediateField F E) f
- simp only [adjoin_simple_le_iff] at this
+ simp only [adjoin_simple_le_iff] at this
exact this hx
#align intermediate_field.exists_finset_of_mem_supr IntermediateField.exists_finset_of_mem_iSup
-/
@@ -999,7 +999,7 @@ variable {K : Type _} [Field K] [Algebra F K]
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
theorem minpoly_gen {α : E} (h : IsIntegral F α) : minpoly F (AdjoinSimple.gen F α) = minpoly F α :=
by
- rw [← adjoin_simple.algebra_map_gen F α] at h
+ rw [← adjoin_simple.algebra_map_gen F α] at h
have inj := (algebraMap F⟮⟯ E).Injective
exact
minpoly.eq_of_algebraMap_eq inj ((isIntegral_algebraMap_iff inj).mp h)
@@ -1085,7 +1085,7 @@ noncomputable def adjoin.powerBasis {x : L} (hx : IsIntegral K x) : PowerBasis K
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
#print IntermediateField.adjoin.finiteDimensional /-
theorem adjoin.finiteDimensional {x : L} (hx : IsIntegral K x) : FiniteDimensional K K⟮⟯ :=
- PowerBasis.finiteDimensional (adjoin.powerBasis hx)
+ PowerBasis.finite (adjoin.powerBasis hx)
#align intermediate_field.adjoin.finite_dimensional IntermediateField.adjoin.finiteDimensional
-/
@@ -1412,7 +1412,7 @@ theorem nonempty_algHom_adjoin_of_splits
(fun s t => x.2.map_hMul ⟨s, _⟩ ⟨t, _⟩) x.2.map_zero (fun s t => x.2.map_add ⟨s, _⟩ ⟨t, _⟩)
x.2.commutes⟩
rcases x.exists_lift_of_splits (hK s hs).1 (hK s hs).2 with ⟨y, h1, h2⟩
- rwa [hx y h1] at h2
+ rwa [hx y h1] at h2
#align intermediate_field.alg_hom_mk_adjoin_splits IntermediateField.nonempty_algHom_adjoin_of_splits
-/
@@ -1421,7 +1421,7 @@ theorem nonempty_algHom_of_adjoin_splits (hS : adjoin F S = ⊤)
(hK : ∀ x ∈ S, IsIntegral F (x : E) ∧ (minpoly F x).Splits (algebraMap F K)) :
Nonempty (E →ₐ[F] K) := by
cases' alg_hom_mk_adjoin_splits hK with ϕ
- rw [hS] at ϕ
+ rw [hS] at ϕ
exact ⟨ϕ.comp top_equiv.symm.to_alg_hom⟩
#align intermediate_field.alg_hom_mk_adjoin_splits' IntermediateField.nonempty_algHom_of_adjoin_splits
-/
@@ -1471,7 +1471,7 @@ instance finiteDimensional_sup [h1 : FiniteDimensional K E1] [h2 : FiniteDimensi
suffices g.range = (E1 ⊔ E2).toSubalgebra
by
have h : FiniteDimensional K g.range.to_submodule := g.to_linear_map.finite_dimensional_range
- rwa [this] at h
+ rwa [this] at h
rw [Algebra.TensorProduct.productMap_range, E1.range_val, E2.range_val, sup_to_subalgebra]
#align intermediate_field.finite_dimensional_sup IntermediateField.finiteDimensional_sup
-/
bump to nightly-2024-01-21-06
mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
Diff
@@ -1438,8 +1438,8 @@ theorem le_sup_toSubalgebra : E1.toSubalgebra ⊔ E2.toSubalgebra ≤ (E1 ⊔ E2
#align intermediate_field.le_sup_to_subalgebra IntermediateField.le_sup_toSubalgebra
-/
-#print IntermediateField.sup_toSubalgebra /-
-theorem sup_toSubalgebra [h1 : FiniteDimensional K E1] [h2 : FiniteDimensional K E2] :
+#print IntermediateField.sup_toSubalgebra_of_left /-
+theorem sup_toSubalgebra_of_left [h1 : FiniteDimensional K E1] [h2 : FiniteDimensional K E2] :
(E1 ⊔ E2).toSubalgebra = E1.toSubalgebra ⊔ E2.toSubalgebra :=
by
let S1 := E1.to_subalgebra
@@ -1460,7 +1460,7 @@ theorem sup_toSubalgebra [h1 : FiniteDimensional K E1] [h2 : FiniteDimensional K
isField_of_isIntegral_of_isField'
(is_integral_sup.mpr ⟨Algebra.isIntegral_of_finite K E1, Algebra.isIntegral_of_finite K E2⟩)
(Field.toIsField K)
-#align intermediate_field.sup_to_subalgebra IntermediateField.sup_toSubalgebra
+#align intermediate_field.sup_to_subalgebra IntermediateField.sup_toSubalgebra_of_left
-/
#print IntermediateField.finiteDimensional_sup /-
bump to nightly-2023-11-29-11
mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
Diff
@@ -1361,7 +1361,7 @@ theorem Lifts.exists_upper_bound (c : Set (Lifts F E K)) (hc : IsChain (· ≤
/-- Extend a lift `x : lifts F E K` to an element `s : E` whose conjugates are all in `K` -/
noncomputable def Lifts.liftOfSplits (x : Lifts F E K) {s : E} (h1 : IsIntegral F s)
(h2 : (minpoly F s).Splits (algebraMap F K)) : Lifts F E K :=
- let h3 : IsIntegral x.1 s := isIntegral_of_isScalarTower h1
+ let h3 : IsIntegral x.1 s := IsIntegral.tower_top h1
let key : (minpoly x.1 s).Splits x.2.toRingHom :=
splits_of_splits_of_dvd _ (map_ne_zero (minpoly.ne_zero h1))
((splits_map_iff _ _).mpr (by convert h2; exact RingHom.ext fun y => x.2.commutes y))
@@ -1520,7 +1520,7 @@ theorem isAlgebraic_iSup {ι : Type _} {f : ι → IntermediateField K L}
rw [is_algebraic_iff, Subtype.coe_mk, ← Subtype.coe_mk x hx, ← is_algebraic_iff]
haveI : ∀ i : Σ i, f i, FiniteDimensional K K⟮⟯ := fun ⟨i, x⟩ =>
adjoin.finite_dimensional (is_integral_iff.1 (isAlgebraic_iff_isIntegral.1 (h i x)))
- apply Algebra.isAlgebraic_of_finite
+ apply Algebra.IsAlgebraic.of_finite
#align intermediate_field.is_algebraic_supr IntermediateField.isAlgebraic_iSup
-/
bump to nightly-2023-11-15-06
mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
Diff
@@ -1267,13 +1267,10 @@ noncomputable instance : OrderBot (Lifts F E K)
noncomputable instance : Inhabited (Lifts F E K) :=
⟨⊥⟩
-#print IntermediateField.Lifts.eq_of_le /-
theorem Lifts.eq_of_le {x y : Lifts F E K} (hxy : x ≤ y) (s : x.1) : x.2 s = y.2 ⟨s, hxy.1 s.Mem⟩ :=
hxy.2 s ⟨s, hxy.1 s.Mem⟩ rfl
#align intermediate_field.lifts.eq_of_le IntermediateField.Lifts.eq_of_le
--/
-#print IntermediateField.Lifts.exists_max_two /-
theorem Lifts.exists_max_two {c : Set (Lifts F E K)} {x y : Lifts F E K} (hc : IsChain (· ≤ ·) c)
(hx : x ∈ Insert.insert ⊥ c) (hy : y ∈ Insert.insert ⊥ c) :
∃ z : Lifts F E K, z ∈ Insert.insert ⊥ c ∧ x ≤ z ∧ y ≤ z :=
@@ -1282,9 +1279,7 @@ theorem Lifts.exists_max_two {c : Set (Lifts F E K)} {x y : Lifts F E K} (hc : I
· exact ⟨y, hy, hxy, le_refl y⟩
· exact ⟨x, hx, le_refl x, hyx⟩
#align intermediate_field.lifts.exists_max_two IntermediateField.Lifts.exists_max_two
--/
-#print IntermediateField.Lifts.exists_max_three /-
theorem Lifts.exists_max_three {c : Set (Lifts F E K)} {x y z : Lifts F E K}
(hc : IsChain (· ≤ ·) c) (hx : x ∈ Insert.insert ⊥ c) (hy : y ∈ Insert.insert ⊥ c)
(hz : z ∈ Insert.insert ⊥ c) :
@@ -1294,9 +1289,7 @@ theorem Lifts.exists_max_three {c : Set (Lifts F E K)} {x y z : Lifts F E K}
obtain ⟨w, hw, hzw, hvw⟩ := lifts.exists_max_two hc hz hv
exact ⟨w, hw, le_trans hxv hvw, le_trans hyv hvw, hzw⟩
#align intermediate_field.lifts.exists_max_three IntermediateField.Lifts.exists_max_three
--/
-#print IntermediateField.Lifts.upperBoundIntermediateField /-
/-- An upper bound on a chain of lifts -/
def Lifts.upperBoundIntermediateField {c : Set (Lifts F E K)} (hc : IsChain (· ≤ ·) c) :
IntermediateField F E
@@ -1316,9 +1309,7 @@ def Lifts.upperBoundIntermediateField {c : Set (Lifts F E K)} (hc : IsChain (·
exact ⟨z, hz, z.1.hMul_mem (hxz.1 ha) (hyz.1 hb)⟩
algebraMap_mem' s := ⟨⊥, Set.mem_insert ⊥ c, algebraMap_mem ⊥ s⟩
#align intermediate_field.lifts.upper_bound_intermediate_field IntermediateField.Lifts.upperBoundIntermediateField
--/
-#print IntermediateField.Lifts.upperBoundAlgHom /-
/-- The lift on the upper bound on a chain of lifts -/
noncomputable def Lifts.upperBoundAlgHom {c : Set (Lifts F E K)} (hc : IsChain (· ≤ ·) c) :
Lifts.upperBoundIntermediateField hc →ₐ[F] K
@@ -1342,14 +1333,11 @@ noncomputable def Lifts.upperBoundAlgHom {c : Set (Lifts F E K)} (hc : IsChain (
rfl
commutes' _ := AlgHom.commutes _ _
#align intermediate_field.lifts.upper_bound_alg_hom IntermediateField.Lifts.upperBoundAlgHom
--/
-#print IntermediateField.Lifts.upperBound /-
/-- An upper bound on a chain of lifts -/
noncomputable def Lifts.upperBound {c : Set (Lifts F E K)} (hc : IsChain (· ≤ ·) c) : Lifts F E K :=
⟨Lifts.upperBoundIntermediateField hc, Lifts.upperBoundAlgHom hc⟩
#align intermediate_field.lifts.upper_bound IntermediateField.Lifts.upperBound
--/
#print IntermediateField.Lifts.exists_upper_bound /-
theorem Lifts.exists_upper_bound (c : Set (Lifts F E K)) (hc : IsChain (· ≤ ·) c) :
@@ -1370,7 +1358,6 @@ theorem Lifts.exists_upper_bound (c : Set (Lifts F E K)) (hc : IsChain (· ≤
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
-#print IntermediateField.Lifts.liftOfSplits /-
/-- Extend a lift `x : lifts F E K` to an element `s : E` whose conjugates are all in `K` -/
noncomputable def Lifts.liftOfSplits (x : Lifts F E K) {s : E} (h1 : IsIntegral F s)
(h2 : (minpoly F s).Splits (algebraMap F K)) : Lifts F E K :=
@@ -1389,11 +1376,9 @@ noncomputable def Lifts.liftOfSplits (x : Lifts F E K) {s : E} (h1 : IsIntegral
simp_rw [mem_roots (map_ne_zero (minpoly.ne_zero h3)), is_root, ← eval₂_eq_eval_map]
exact map_root_of_splits x.2.toRingHom key (ne_of_gt (minpoly.degree_pos h3))⟩⟩⟩
#align intermediate_field.lifts.lift_of_splits IntermediateField.Lifts.liftOfSplits
--/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
-#print IntermediateField.Lifts.le_lifts_of_splits /-
theorem Lifts.le_lifts_of_splits (x : Lifts F E K) {s : E} (h1 : IsIntegral F s)
(h2 : (minpoly F s).Splits (algebraMap F K)) : x ≤ x.lift_of_splits h1 h2 :=
⟨fun z hz => algebraMap_mem x.1⟮⟯ ⟨z, hz⟩, fun t u htu =>
@@ -1403,14 +1388,11 @@ theorem Lifts.le_lifts_of_splits (x : Lifts F E K) {s : E} (h1 : IsIntegral F s)
letI : Algebra x.1 K := x.2.toRingHom.toAlgebra
exact AlgHom.commutes _ t)⟩
#align intermediate_field.lifts.le_lifts_of_splits IntermediateField.Lifts.le_lifts_of_splits
--/
-#print IntermediateField.Lifts.mem_lifts_of_splits /-
theorem Lifts.mem_lifts_of_splits (x : Lifts F E K) {s : E} (h1 : IsIntegral F s)
(h2 : (minpoly F s).Splits (algebraMap F K)) : s ∈ (x.lift_of_splits h1 h2).1 :=
mem_adjoin_simple_self x.1 s
#align intermediate_field.lifts.mem_lifts_of_splits IntermediateField.Lifts.mem_lifts_of_splits
--/
#print IntermediateField.Lifts.exists_lift_of_splits /-
theorem Lifts.exists_lift_of_splits (x : Lifts F E K) {s : E} (h1 : IsIntegral F s)
@@ -1419,8 +1401,8 @@ theorem Lifts.exists_lift_of_splits (x : Lifts F E K) {s : E} (h1 : IsIntegral F
#align intermediate_field.lifts.exists_lift_of_splits IntermediateField.Lifts.exists_lift_of_splits
-/
-#print IntermediateField.algHom_mk_adjoin_splits /-
-theorem algHom_mk_adjoin_splits
+#print IntermediateField.nonempty_algHom_adjoin_of_splits /-
+theorem nonempty_algHom_adjoin_of_splits
(hK : ∀ s ∈ S, IsIntegral F (s : E) ∧ (minpoly F s).Splits (algebraMap F K)) :
Nonempty (adjoin F S →ₐ[F] K) :=
by
@@ -1431,17 +1413,17 @@ theorem algHom_mk_adjoin_splits
x.2.commutes⟩
rcases x.exists_lift_of_splits (hK s hs).1 (hK s hs).2 with ⟨y, h1, h2⟩
rwa [hx y h1] at h2
-#align intermediate_field.alg_hom_mk_adjoin_splits IntermediateField.algHom_mk_adjoin_splits
+#align intermediate_field.alg_hom_mk_adjoin_splits IntermediateField.nonempty_algHom_adjoin_of_splits
-/
-#print IntermediateField.algHom_mk_adjoin_splits' /-
-theorem algHom_mk_adjoin_splits' (hS : adjoin F S = ⊤)
+#print IntermediateField.nonempty_algHom_of_adjoin_splits /-
+theorem nonempty_algHom_of_adjoin_splits (hS : adjoin F S = ⊤)
(hK : ∀ x ∈ S, IsIntegral F (x : E) ∧ (minpoly F x).Splits (algebraMap F K)) :
Nonempty (E →ₐ[F] K) := by
cases' alg_hom_mk_adjoin_splits hK with ϕ
rw [hS] at ϕ
exact ⟨ϕ.comp top_equiv.symm.to_alg_hom⟩
-#align intermediate_field.alg_hom_mk_adjoin_splits' IntermediateField.algHom_mk_adjoin_splits'
+#align intermediate_field.alg_hom_mk_adjoin_splits' IntermediateField.nonempty_algHom_of_adjoin_splits
-/
end AlgHomMkAdjoinSplits
bump to nightly-2023-10-29-06
mathlib commit https://github.com/leanprover-community/mathlib/commit/3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe
Diff
@@ -330,33 +330,33 @@ theorem restrictScalars_top {K : Type _} [Field K] [Algebra K E] [Algebra K F]
#align intermediate_field.restrict_scalars_top IntermediateField.restrictScalars_top
-/
-#print IntermediateField.AlgHom.fieldRange_eq_map /-
-theorem IntermediateField.AlgHom.fieldRange_eq_map {K : Type _} [Field K] [Algebra F K]
- (f : E →ₐ[F] K) : f.fieldRange = IntermediateField.map f ⊤ :=
+#print AlgHom.fieldRange_eq_map /-
+theorem AlgHom.fieldRange_eq_map {K : Type _} [Field K] [Algebra F K] (f : E →ₐ[F] K) :
+ f.fieldRange = IntermediateField.map f ⊤ :=
SetLike.ext' Set.image_univ.symm
-#align alg_hom.field_range_eq_map IntermediateField.AlgHom.fieldRange_eq_map
+#align alg_hom.field_range_eq_map AlgHom.fieldRange_eq_map
-/
-#print IntermediateField.AlgHom.map_fieldRange /-
-theorem IntermediateField.AlgHom.map_fieldRange {K L : Type _} [Field K] [Field L] [Algebra F K]
- [Algebra F L] (f : E →ₐ[F] K) (g : K →ₐ[F] L) : f.fieldRange.map g = (g.comp f).fieldRange :=
+#print AlgHom.map_fieldRange /-
+theorem AlgHom.map_fieldRange {K L : Type _} [Field K] [Field L] [Algebra F K] [Algebra F L]
+ (f : E →ₐ[F] K) (g : K →ₐ[F] L) : f.fieldRange.map g = (g.comp f).fieldRange :=
SetLike.ext' (Set.range_comp g f).symm
-#align alg_hom.map_field_range IntermediateField.AlgHom.map_fieldRange
+#align alg_hom.map_field_range AlgHom.map_fieldRange
-/
-#print IntermediateField.AlgHom.fieldRange_eq_top /-
-theorem IntermediateField.AlgHom.fieldRange_eq_top {K : Type _} [Field K] [Algebra F K]
- {f : E →ₐ[F] K} : f.fieldRange = ⊤ ↔ Function.Surjective f :=
+#print AlgHom.fieldRange_eq_top /-
+theorem AlgHom.fieldRange_eq_top {K : Type _} [Field K] [Algebra F K] {f : E →ₐ[F] K} :
+ f.fieldRange = ⊤ ↔ Function.Surjective f :=
SetLike.ext'_iff.trans Set.range_iff_surjective
-#align alg_hom.field_range_eq_top IntermediateField.AlgHom.fieldRange_eq_top
+#align alg_hom.field_range_eq_top AlgHom.fieldRange_eq_top
-/
-#print IntermediateField.AlgEquiv.fieldRange_eq_top /-
+#print AlgEquiv.fieldRange_eq_top /-
@[simp]
-theorem IntermediateField.AlgEquiv.fieldRange_eq_top {K : Type _} [Field K] [Algebra F K]
- (f : E ≃ₐ[F] K) : (f : E →ₐ[F] K).fieldRange = ⊤ :=
- IntermediateField.AlgHom.fieldRange_eq_top.mpr f.Surjective
-#align alg_equiv.field_range_eq_top IntermediateField.AlgEquiv.fieldRange_eq_top
+theorem AlgEquiv.fieldRange_eq_top {K : Type _} [Field K] [Algebra F K] (f : E ≃ₐ[F] K) :
+ (f : E →ₐ[F] K).fieldRange = ⊤ :=
+ AlgHom.fieldRange_eq_top.mpr f.Surjective
+#align alg_equiv.field_range_eq_top AlgEquiv.fieldRange_eq_top
-/
end Lattice
bump to nightly-2023-09-24-06
mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451
Diff
@@ -3,10 +3,10 @@ Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
-import Mathbin.FieldTheory.IntermediateField
-import Mathbin.FieldTheory.Separable
-import Mathbin.FieldTheory.SplittingField.IsSplittingField
-import Mathbin.RingTheory.TensorProduct
+import FieldTheory.IntermediateField
+import FieldTheory.Separable
+import FieldTheory.SplittingField.IsSplittingField
+import RingTheory.TensorProduct
#align_import field_theory.adjoin from "leanprover-community/mathlib"@"6b31d1eebd64eab86d5bd9936bfaada6ca8b5842"
bump to nightly-2023-09-20-06
mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451
Diff
@@ -997,7 +997,6 @@ variable {F : Type _} [Field F] {E : Type _} [Field E] [Algebra F E] {α : E}
variable {K : Type _} [Field K] [Algebra F K]
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
-#print IntermediateField.minpoly_gen /-
theorem minpoly_gen {α : E} (h : IsIntegral F α) : minpoly F (AdjoinSimple.gen F α) = minpoly F α :=
by
rw [← adjoin_simple.algebra_map_gen F α] at h
@@ -1005,8 +1004,7 @@ theorem minpoly_gen {α : E} (h : IsIntegral F α) : minpoly F (AdjoinSimple.gen
exact
minpoly.eq_of_algebraMap_eq inj ((isIntegral_algebraMap_iff inj).mp h)
(adjoin_simple.algebra_map_gen _ _).symm
-#align intermediate_field.minpoly_gen IntermediateField.minpoly_gen
--/
+#align intermediate_field.minpoly_gen IntermediateField.minpoly_genₓ
variable (F)
bump to nightly-2023-08-17-09
mathlib commit https://github.com/leanprover-community/mathlib/commit/32a7e535287f9c73f2e4d2aef306a39190f0b504
Diff
@@ -727,7 +727,7 @@ theorem adjoin_simple_isCompactElement (x : E) : IsCompactElement F⟮⟯ :=
neg_mem' := by
rintro x ⟨-, ⟨E, rfl⟩, ⟨-, ⟨hE, rfl⟩, hx⟩⟩
exact mem_Union_of_mem E (mem_Union_of_mem hE (E.neg_mem hx))
- mul_mem' :=
+ hMul_mem' :=
by
rintro x₁ x₂ ⟨-, ⟨F₁, rfl⟩, ⟨-, ⟨hF₁, rfl⟩, hx₁⟩⟩ ⟨-, ⟨F₂, rfl⟩, ⟨-, ⟨hF₂, rfl⟩, hx₂⟩⟩
obtain ⟨F₃, hF₃, h₁₃, h₂₃⟩ := hs F₁ hF₁ F₂ hF₂
@@ -1312,10 +1312,10 @@ def Lifts.upperBoundIntermediateField {c : Set (Lifts F E K)} (hc : IsChain (·
rintro _ _ ⟨x, hx, ha⟩ ⟨y, hy, hb⟩
obtain ⟨z, hz, hxz, hyz⟩ := lifts.exists_max_two hc hx hy
exact ⟨z, hz, z.1.add_mem (hxz.1 ha) (hyz.1 hb)⟩
- mul_mem' := by
+ hMul_mem' := by
rintro _ _ ⟨x, hx, ha⟩ ⟨y, hy, hb⟩
obtain ⟨z, hz, hxz, hyz⟩ := lifts.exists_max_two hc hx hy
- exact ⟨z, hz, z.1.mul_mem (hxz.1 ha) (hyz.1 hb)⟩
+ exact ⟨z, hz, z.1.hMul_mem (hxz.1 ha) (hyz.1 hb)⟩
algebraMap_mem' s := ⟨⊥, Set.mem_insert ⊥ c, algebraMap_mem ⊥ s⟩
#align intermediate_field.lifts.upper_bound_intermediate_field IntermediateField.Lifts.upperBoundIntermediateField
-/
@@ -1340,7 +1340,7 @@ noncomputable def Lifts.upperBoundAlgHom {c : Set (Lifts F E K)} (hc : IsChain (
obtain ⟨w, hw, hxw, hyw, hzw⟩ :=
lifts.exists_max_three hc (Classical.choose_spec s.mem).1 (Classical.choose_spec t.mem).1
(Classical.choose_spec (s * t).Mem).1
- rw [lifts.eq_of_le hxw, lifts.eq_of_le hyw, lifts.eq_of_le hzw, ← w.2.map_mul]
+ rw [lifts.eq_of_le hxw, lifts.eq_of_le hyw, lifts.eq_of_le hzw, ← w.2.map_hMul]
rfl
commutes' _ := AlgHom.commutes _ _
#align intermediate_field.lifts.upper_bound_alg_hom IntermediateField.Lifts.upperBoundAlgHom
@@ -1429,7 +1429,7 @@ theorem algHom_mk_adjoin_splits
obtain ⟨x : lifts F E K, hx⟩ := zorn_partialOrder lifts.exists_upper_bound
refine'
⟨AlgHom.mk (fun s => x.2 ⟨s, adjoin_le_iff.mpr (fun s hs => _) s.Mem⟩) x.2.map_one
- (fun s t => x.2.map_mul ⟨s, _⟩ ⟨t, _⟩) x.2.map_zero (fun s t => x.2.map_add ⟨s, _⟩ ⟨t, _⟩)
+ (fun s t => x.2.map_hMul ⟨s, _⟩ ⟨t, _⟩) x.2.map_zero (fun s t => x.2.map_add ⟨s, _⟩ ⟨t, _⟩)
x.2.commutes⟩
rcases x.exists_lift_of_splits (hK s hs).1 (hK s hs).2 with ⟨y, h1, h2⟩
rwa [hx y h1] at h2
bump to nightly-2023-07-27-09
mathlib commit https://github.com/leanprover-community/mathlib/commit/d11f435d4e34a6cea0a1797d6b625b0c170be845
Diff
@@ -1102,18 +1102,18 @@ theorem adjoin.finrank {x : L} (hx : IsIntegral K x) :
-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
-#print IntermediateField.minpoly.natDegree_le /-
-theorem IntermediateField.minpoly.natDegree_le {x : L} [FiniteDimensional K L]
- (hx : IsIntegral K x) : (minpoly K x).natDegree ≤ finrank K L :=
+#print minpoly.natDegree_le /-
+theorem minpoly.natDegree_le {x : L} [FiniteDimensional K L] (hx : IsIntegral K x) :
+ (minpoly K x).natDegree ≤ finrank K L :=
le_of_eq_of_le (IntermediateField.adjoin.finrank hx).symm K⟮⟯.toSubmodule.finrank_le
-#align minpoly.nat_degree_le IntermediateField.minpoly.natDegree_le
+#align minpoly.nat_degree_le minpoly.natDegree_le
-/
-#print IntermediateField.minpoly.degree_le /-
-theorem IntermediateField.minpoly.degree_le {x : L} [FiniteDimensional K L] (hx : IsIntegral K x) :
+#print minpoly.degree_le /-
+theorem minpoly.degree_le {x : L} [FiniteDimensional K L] (hx : IsIntegral K x) :
(minpoly K x).degree ≤ finrank K L :=
- degree_le_of_natDegree_le (IntermediateField.minpoly.natDegree_le hx)
-#align minpoly.degree_le IntermediateField.minpoly.degree_le
+ degree_le_of_natDegree_le (minpoly.natDegree_le hx)
+#align minpoly.degree_le minpoly.degree_le
-/
end PowerBasis
bump to nightly-2023-07-20-09
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
Diff
@@ -2,17 +2,14 @@
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-
-! This file was ported from Lean 3 source module field_theory.adjoin
-! leanprover-community/mathlib commit 6b31d1eebd64eab86d5bd9936bfaada6ca8b5842
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathbin.FieldTheory.IntermediateField
import Mathbin.FieldTheory.Separable
import Mathbin.FieldTheory.SplittingField.IsSplittingField
import Mathbin.RingTheory.TensorProduct
+#align_import field_theory.adjoin from "leanprover-community/mathlib"@"6b31d1eebd64eab86d5bd9936bfaada6ca8b5842"
+
/-!
# Adjoining Elements to Fields
bump to nightly-2023-06-28-03
mathlib commit https://github.com/leanprover-community/mathlib/commit/8b981918a93bc45a8600de608cde7944a80d92b9
Diff
@@ -487,7 +487,7 @@ theorem adjoin_insert_adjoin (x : E) :
adjoin F (insert x (adjoin F S : Set E)) = adjoin F (insert x S) :=
le_antisymm
(adjoin_le_iff.mpr
- (Set.insert_subset.mpr
+ (Set.insert_subset_iff.mpr
⟨subset_adjoin _ _ (Set.mem_insert _ _),
adjoin_le_iff.mpr (subset_adjoin_of_subset_right _ _ (Set.subset_insert _ _))⟩))
(adjoin.mono _ _ _ (Set.insert_subset_insert (subset_adjoin _ _)))
bump to nightly-2023-06-18-23
mathlib commit https://github.com/leanprover-community/mathlib/commit/2a0ce625dbb0ffbc7d1316597de0b25c1ec75303
Diff
@@ -562,7 +562,6 @@ theorem adjoin_induction {s : Set E} {p : E → Prop} {x} (h : x ∈ adjoin F s)
#align intermediate_field.adjoin_induction IntermediateField.adjoin_induction
-/
-#print IntermediateField.Insert /-
--this definition of notation is courtesy of Kyle Miller on zulip
/-- Variation on `set.insert` to enable good notation for adjoining elements to fields.
Used to preferentially use `singleton` rather than `insert` when adjoining one element.
@@ -570,19 +569,14 @@ Used to preferentially use `singleton` rather than `insert` when adjoining one e
class Insert {α : Type _} (s : Set α) where
insert : α → Set α
#align intermediate_field.insert IntermediateField.Insert
--/
-#print IntermediateField.insertEmpty /-
instance (priority := 1000) insertEmpty {α : Type _} : Insert (∅ : Set α)
where insert x := @singleton _ _ Set.hasSingleton x
#align intermediate_field.insert_empty IntermediateField.insertEmpty
--/
-#print IntermediateField.insertNonempty /-
instance (priority := 900) insertNonempty {α : Type _} (s : Set α) : Insert s
where insert x := Insert.insert x s
#align intermediate_field.insert_nonempty IntermediateField.insertNonempty
--/
notation3:max K"⟮"(l ", "* => foldr (h t => Insert.insert t h) ∅)"⟯" => adjoin K l
bump to nightly-2023-06-13-21
mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423
Diff
@@ -59,17 +59,22 @@ section Lattice
variable {F : Type _} [Field F] {E : Type _} [Field E] [Algebra F E]
+#print IntermediateField.adjoin_le_iff /-
@[simp]
theorem adjoin_le_iff {S : Set E} {T : IntermediateField F E} : adjoin F S ≤ T ↔ S ≤ T :=
⟨fun H => le_trans (le_trans (Set.subset_union_right _ _) Subfield.subset_closure) H, fun H =>
(@Subfield.closure_le E _ (Set.range (algebraMap F E) ∪ S) T.toSubfield).mpr
(Set.union_subset (IntermediateField.set_range_subset T) H)⟩
#align intermediate_field.adjoin_le_iff IntermediateField.adjoin_le_iff
+-/
+#print IntermediateField.gc /-
theorem gc : GaloisConnection (adjoin F : Set E → IntermediateField F E) coe := fun _ _ =>
adjoin_le_iff
#align intermediate_field.gc IntermediateField.gc
+-/
+#print IntermediateField.gi /-
/-- Galois insertion between `adjoin` and `coe`. -/
def gi : GaloisInsertion (adjoin F : Set E → IntermediateField F E) coe
where
@@ -78,6 +83,7 @@ def gi : GaloisInsertion (adjoin F : Set E → IntermediateField F E) coe
le_l_u S := (IntermediateField.gc (S : Set E) (adjoin F S)).1 <| le_rfl
choice_eq _ _ := copy_eq _ _ _
#align intermediate_field.gi IntermediateField.gi
+-/
instance : CompleteLattice (IntermediateField F E) :=
GaloisInsertion.liftCompleteLattice IntermediateField.gi
@@ -85,97 +91,132 @@ instance : CompleteLattice (IntermediateField F E) :=
instance : Inhabited (IntermediateField F E) :=
⟨⊤⟩
+#print IntermediateField.coe_bot /-
theorem coe_bot : ↑(⊥ : IntermediateField F E) = Set.range (algebraMap F E) :=
by
change ↑(Subfield.closure (Set.range (algebraMap F E) ∪ ∅)) = Set.range (algebraMap F E)
simp [← Set.image_univ, ← RingHom.map_field_closure]
#align intermediate_field.coe_bot IntermediateField.coe_bot
+-/
+#print IntermediateField.mem_bot /-
theorem mem_bot {x : E} : x ∈ (⊥ : IntermediateField F E) ↔ x ∈ Set.range (algebraMap F E) :=
Set.ext_iff.mp coe_bot x
#align intermediate_field.mem_bot IntermediateField.mem_bot
+-/
+#print IntermediateField.bot_toSubalgebra /-
@[simp]
theorem bot_toSubalgebra : (⊥ : IntermediateField F E).toSubalgebra = ⊥ := by ext;
rw [mem_to_subalgebra, Algebra.mem_bot, mem_bot]
#align intermediate_field.bot_to_subalgebra IntermediateField.bot_toSubalgebra
+-/
+#print IntermediateField.coe_top /-
@[simp]
theorem coe_top : ↑(⊤ : IntermediateField F E) = (Set.univ : Set E) :=
rfl
#align intermediate_field.coe_top IntermediateField.coe_top
+-/
+#print IntermediateField.mem_top /-
@[simp]
theorem mem_top {x : E} : x ∈ (⊤ : IntermediateField F E) :=
trivial
#align intermediate_field.mem_top IntermediateField.mem_top
+-/
+#print IntermediateField.top_toSubalgebra /-
@[simp]
theorem top_toSubalgebra : (⊤ : IntermediateField F E).toSubalgebra = ⊤ :=
rfl
#align intermediate_field.top_to_subalgebra IntermediateField.top_toSubalgebra
+-/
+#print IntermediateField.top_toSubfield /-
@[simp]
theorem top_toSubfield : (⊤ : IntermediateField F E).toSubfield = ⊤ :=
rfl
#align intermediate_field.top_to_subfield IntermediateField.top_toSubfield
+-/
+#print IntermediateField.coe_inf /-
@[simp, norm_cast]
theorem coe_inf (S T : IntermediateField F E) : (↑(S ⊓ T) : Set E) = S ∩ T :=
rfl
#align intermediate_field.coe_inf IntermediateField.coe_inf
+-/
+#print IntermediateField.mem_inf /-
@[simp]
theorem mem_inf {S T : IntermediateField F E} {x : E} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=
Iff.rfl
#align intermediate_field.mem_inf IntermediateField.mem_inf
+-/
+#print IntermediateField.inf_toSubalgebra /-
@[simp]
theorem inf_toSubalgebra (S T : IntermediateField F E) :
(S ⊓ T).toSubalgebra = S.toSubalgebra ⊓ T.toSubalgebra :=
rfl
#align intermediate_field.inf_to_subalgebra IntermediateField.inf_toSubalgebra
+-/
+#print IntermediateField.inf_toSubfield /-
@[simp]
theorem inf_toSubfield (S T : IntermediateField F E) :
(S ⊓ T).toSubfield = S.toSubfield ⊓ T.toSubfield :=
rfl
#align intermediate_field.inf_to_subfield IntermediateField.inf_toSubfield
+-/
+#print IntermediateField.coe_sInf /-
@[simp, norm_cast]
theorem coe_sInf (S : Set (IntermediateField F E)) : (↑(sInf S) : Set E) = sInf (coe '' S) :=
rfl
#align intermediate_field.coe_Inf IntermediateField.coe_sInf
+-/
+#print IntermediateField.sInf_toSubalgebra /-
@[simp]
theorem sInf_toSubalgebra (S : Set (IntermediateField F E)) :
(sInf S).toSubalgebra = sInf (toSubalgebra '' S) :=
SetLike.coe_injective <| by simp [Set.sUnion_image]
#align intermediate_field.Inf_to_subalgebra IntermediateField.sInf_toSubalgebra
+-/
+#print IntermediateField.sInf_toSubfield /-
@[simp]
theorem sInf_toSubfield (S : Set (IntermediateField F E)) :
(sInf S).toSubfield = sInf (toSubfield '' S) :=
SetLike.coe_injective <| by simp [Set.sUnion_image]
#align intermediate_field.Inf_to_subfield IntermediateField.sInf_toSubfield
+-/
+#print IntermediateField.coe_iInf /-
@[simp, norm_cast]
theorem coe_iInf {ι : Sort _} (S : ι → IntermediateField F E) : (↑(iInf S) : Set E) = ⋂ i, S i := by
simp [iInf]
#align intermediate_field.coe_infi IntermediateField.coe_iInf
+-/
+#print IntermediateField.iInf_toSubalgebra /-
@[simp]
theorem iInf_toSubalgebra {ι : Sort _} (S : ι → IntermediateField F E) :
(iInf S).toSubalgebra = ⨅ i, (S i).toSubalgebra :=
SetLike.coe_injective <| by simp [iInf]
#align intermediate_field.infi_to_subalgebra IntermediateField.iInf_toSubalgebra
+-/
+#print IntermediateField.iInf_toSubfield /-
@[simp]
theorem iInf_toSubfield {ι : Sort _} (S : ι → IntermediateField F E) :
(iInf S).toSubfield = ⨅ i, (S i).toSubfield :=
SetLike.coe_injective <| by simp [iInf]
#align intermediate_field.infi_to_subfield IntermediateField.iInf_toSubfield
+-/
+#print IntermediateField.equivOfEq /-
/-- Construct an algebra isomorphism from an equality of intermediate fields -/
@[simps apply]
def equivOfEq {S T : IntermediateField F E} (h : S = T) : S ≃ₐ[F] T := by
@@ -184,53 +225,71 @@ def equivOfEq {S T : IntermediateField F E} (h : S = T) : S ≃ₐ[F] T := by
invFun := fun x => ⟨x, _⟩ .. } <;>
tidy
#align intermediate_field.equiv_of_eq IntermediateField.equivOfEq
+-/
+#print IntermediateField.equivOfEq_symm /-
@[simp]
theorem equivOfEq_symm {S T : IntermediateField F E} (h : S = T) :
(equivOfEq h).symm = equivOfEq h.symm :=
rfl
#align intermediate_field.equiv_of_eq_symm IntermediateField.equivOfEq_symm
+-/
+#print IntermediateField.equivOfEq_rfl /-
@[simp]
theorem equivOfEq_rfl (S : IntermediateField F E) : equivOfEq (rfl : S = S) = AlgEquiv.refl := by
ext; rfl
#align intermediate_field.equiv_of_eq_rfl IntermediateField.equivOfEq_rfl
+-/
+#print IntermediateField.equivOfEq_trans /-
@[simp]
theorem equivOfEq_trans {S T U : IntermediateField F E} (hST : S = T) (hTU : T = U) :
(equivOfEq hST).trans (equivOfEq hTU) = equivOfEq (trans hST hTU) :=
rfl
#align intermediate_field.equiv_of_eq_trans IntermediateField.equivOfEq_trans
+-/
variable (F E)
+#print IntermediateField.botEquiv /-
/-- The bottom intermediate_field is isomorphic to the field. -/
noncomputable def botEquiv : (⊥ : IntermediateField F E) ≃ₐ[F] F :=
(Subalgebra.equivOfEq _ _ bot_toSubalgebra).trans (Algebra.botEquiv F E)
#align intermediate_field.bot_equiv IntermediateField.botEquiv
+-/
variable {F E}
+#print IntermediateField.botEquiv_def /-
@[simp]
theorem botEquiv_def (x : F) : botEquiv F E (algebraMap F (⊥ : IntermediateField F E) x) = x :=
AlgEquiv.commutes (botEquiv F E) x
#align intermediate_field.bot_equiv_def IntermediateField.botEquiv_def
+-/
+#print IntermediateField.botEquiv_symm /-
@[simp]
theorem botEquiv_symm (x : F) : (botEquiv F E).symm x = algebraMap F _ x :=
rfl
#align intermediate_field.bot_equiv_symm IntermediateField.botEquiv_symm
+-/
+#print IntermediateField.algebraOverBot /-
noncomputable instance algebraOverBot : Algebra (⊥ : IntermediateField F E) F :=
(IntermediateField.botEquiv F E).toAlgHom.toRingHom.toAlgebra
#align intermediate_field.algebra_over_bot IntermediateField.algebraOverBot
+-/
+#print IntermediateField.coe_algebraMap_over_bot /-
theorem coe_algebraMap_over_bot :
(algebraMap (⊥ : IntermediateField F E) F : (⊥ : IntermediateField F E) → F) =
IntermediateField.botEquiv F E :=
rfl
#align intermediate_field.coe_algebra_map_over_bot IntermediateField.coe_algebraMap_over_bot
+-/
+#print IntermediateField.isScalarTower_over_bot /-
instance isScalarTower_over_bot : IsScalarTower (⊥ : IntermediateField F E) F E :=
IsScalarTower.of_algebraMap_eq
(by
@@ -239,7 +298,9 @@ instance isScalarTower_over_bot : IsScalarTower (⊥ : IntermediateField F E) F
rw [coe_algebra_map_over_bot, (bot_equiv F E).apply_symm_apply, bot_equiv_symm,
IsScalarTower.algebraMap_apply F (⊥ : IntermediateField F E) E])
#align intermediate_field.is_scalar_tower_over_bot IntermediateField.isScalarTower_over_bot
+-/
+#print IntermediateField.topEquiv /-
/-- The top intermediate_field is isomorphic to the field.
This is the intermediate field version of `subalgebra.top_equiv`. -/
@@ -247,44 +308,59 @@ This is the intermediate field version of `subalgebra.top_equiv`. -/
def topEquiv : (⊤ : IntermediateField F E) ≃ₐ[F] E :=
(Subalgebra.equivOfEq _ _ top_toSubalgebra).trans Subalgebra.topEquiv
#align intermediate_field.top_equiv IntermediateField.topEquiv
+-/
+#print IntermediateField.topEquiv_symm_apply_coe /-
@[simp]
theorem topEquiv_symm_apply_coe (a : E) : ↑(topEquiv.symm a : (⊤ : IntermediateField F E)) = a :=
rfl
#align intermediate_field.top_equiv_symm_apply_coe IntermediateField.topEquiv_symm_apply_coe
+-/
+#print IntermediateField.restrictScalars_bot_eq_self /-
@[simp]
theorem restrictScalars_bot_eq_self (K : IntermediateField F E) :
(⊥ : IntermediateField K E).restrictScalars _ = K := by ext; rw [mem_restrict_scalars, mem_bot];
exact set.ext_iff.mp Subtype.range_coe x
#align intermediate_field.restrict_scalars_bot_eq_self IntermediateField.restrictScalars_bot_eq_self
+-/
+#print IntermediateField.restrictScalars_top /-
@[simp]
theorem restrictScalars_top {K : Type _} [Field K] [Algebra K E] [Algebra K F]
[IsScalarTower K F E] : (⊤ : IntermediateField F E).restrictScalars K = ⊤ :=
rfl
#align intermediate_field.restrict_scalars_top IntermediateField.restrictScalars_top
+-/
+#print IntermediateField.AlgHom.fieldRange_eq_map /-
theorem IntermediateField.AlgHom.fieldRange_eq_map {K : Type _} [Field K] [Algebra F K]
(f : E →ₐ[F] K) : f.fieldRange = IntermediateField.map f ⊤ :=
SetLike.ext' Set.image_univ.symm
#align alg_hom.field_range_eq_map IntermediateField.AlgHom.fieldRange_eq_map
+-/
+#print IntermediateField.AlgHom.map_fieldRange /-
theorem IntermediateField.AlgHom.map_fieldRange {K L : Type _} [Field K] [Field L] [Algebra F K]
[Algebra F L] (f : E →ₐ[F] K) (g : K →ₐ[F] L) : f.fieldRange.map g = (g.comp f).fieldRange :=
SetLike.ext' (Set.range_comp g f).symm
#align alg_hom.map_field_range IntermediateField.AlgHom.map_fieldRange
+-/
+#print IntermediateField.AlgHom.fieldRange_eq_top /-
theorem IntermediateField.AlgHom.fieldRange_eq_top {K : Type _} [Field K] [Algebra F K]
{f : E →ₐ[F] K} : f.fieldRange = ⊤ ↔ Function.Surjective f :=
SetLike.ext'_iff.trans Set.range_iff_surjective
#align alg_hom.field_range_eq_top IntermediateField.AlgHom.fieldRange_eq_top
+-/
+#print IntermediateField.AlgEquiv.fieldRange_eq_top /-
@[simp]
theorem IntermediateField.AlgEquiv.fieldRange_eq_top {K : Type _} [Field K] [Algebra F K]
(f : E ≃ₐ[F] K) : (f : E →ₐ[F] K).fieldRange = ⊤ :=
IntermediateField.AlgHom.fieldRange_eq_top.mpr f.Surjective
#align alg_equiv.field_range_eq_top IntermediateField.AlgEquiv.fieldRange_eq_top
+-/
end Lattice
@@ -292,15 +368,20 @@ section AdjoinDef
variable (F : Type _) [Field F] {E : Type _} [Field E] [Algebra F E] (S : Set E)
+#print IntermediateField.adjoin_eq_range_algebraMap_adjoin /-
theorem adjoin_eq_range_algebraMap_adjoin :
(adjoin F S : Set E) = Set.range (algebraMap (adjoin F S) E) :=
Subtype.range_coe.symm
#align intermediate_field.adjoin_eq_range_algebra_map_adjoin IntermediateField.adjoin_eq_range_algebraMap_adjoin
+-/
+#print IntermediateField.adjoin.algebraMap_mem /-
theorem adjoin.algebraMap_mem (x : F) : algebraMap F E x ∈ adjoin F S :=
IntermediateField.algebraMap_mem (adjoin F S) x
#align intermediate_field.adjoin.algebra_map_mem IntermediateField.adjoin.algebraMap_mem
+-/
+#print IntermediateField.adjoin.range_algebraMap_subset /-
theorem adjoin.range_algebraMap_subset : Set.range (algebraMap F E) ⊆ adjoin F S :=
by
intro x hx
@@ -308,45 +389,65 @@ theorem adjoin.range_algebraMap_subset : Set.range (algebraMap F E) ⊆ adjoin F
rw [← hf]
exact adjoin.algebra_map_mem F S f
#align intermediate_field.adjoin.range_algebra_map_subset IntermediateField.adjoin.range_algebraMap_subset
+-/
+#print IntermediateField.adjoin.fieldCoe /-
instance adjoin.fieldCoe : CoeTC F (adjoin F S)
where coe x := ⟨algebraMap F E x, adjoin.algebraMap_mem F S x⟩
#align intermediate_field.adjoin.field_coe IntermediateField.adjoin.fieldCoe
+-/
+#print IntermediateField.subset_adjoin /-
theorem subset_adjoin : S ⊆ adjoin F S := fun x hx => Subfield.subset_closure (Or.inr hx)
#align intermediate_field.subset_adjoin IntermediateField.subset_adjoin
+-/
+#print IntermediateField.adjoin.setCoe /-
instance adjoin.setCoe : CoeTC S (adjoin F S) where coe x := ⟨x, subset_adjoin F S (Subtype.mem x)⟩
#align intermediate_field.adjoin.set_coe IntermediateField.adjoin.setCoe
+-/
+#print IntermediateField.adjoin.mono /-
@[mono]
theorem adjoin.mono (T : Set E) (h : S ⊆ T) : adjoin F S ≤ adjoin F T :=
GaloisConnection.monotone_l gc h
#align intermediate_field.adjoin.mono IntermediateField.adjoin.mono
+-/
+#print IntermediateField.adjoin_contains_field_as_subfield /-
theorem adjoin_contains_field_as_subfield (F : Subfield E) : (F : Set E) ⊆ adjoin F S := fun x hx =>
adjoin.algebraMap_mem F S ⟨x, hx⟩
#align intermediate_field.adjoin_contains_field_as_subfield IntermediateField.adjoin_contains_field_as_subfield
+-/
+#print IntermediateField.subset_adjoin_of_subset_left /-
theorem subset_adjoin_of_subset_left {F : Subfield E} {T : Set E} (HT : T ⊆ F) : T ⊆ adjoin F S :=
fun x hx => (adjoin F S).algebraMap_mem ⟨x, HT hx⟩
#align intermediate_field.subset_adjoin_of_subset_left IntermediateField.subset_adjoin_of_subset_left
+-/
+#print IntermediateField.subset_adjoin_of_subset_right /-
theorem subset_adjoin_of_subset_right {T : Set E} (H : T ⊆ S) : T ⊆ adjoin F S := fun x hx =>
subset_adjoin F S (H hx)
#align intermediate_field.subset_adjoin_of_subset_right IntermediateField.subset_adjoin_of_subset_right
+-/
+#print IntermediateField.adjoin_empty /-
@[simp]
theorem adjoin_empty (F E : Type _) [Field F] [Field E] [Algebra F E] : adjoin F (∅ : Set E) = ⊥ :=
eq_bot_iff.mpr (adjoin_le_iff.mpr (Set.empty_subset _))
#align intermediate_field.adjoin_empty IntermediateField.adjoin_empty
+-/
+#print IntermediateField.adjoin_univ /-
@[simp]
theorem adjoin_univ (F E : Type _) [Field F] [Field E] [Algebra F E] :
adjoin F (Set.univ : Set E) = ⊤ :=
eq_top_iff.mpr <| subset_adjoin _ _
#align intermediate_field.adjoin_univ IntermediateField.adjoin_univ
+-/
+#print IntermediateField.adjoin_le_subfield /-
/-- If `K` is a field with `F ⊆ K` and `S ⊆ K` then `adjoin F S ≤ K`. -/
theorem adjoin_le_subfield {K : Subfield E} (HF : Set.range (algebraMap F E) ⊆ K) (HS : S ⊆ K) :
(adjoin F S).toSubfield ≤ K := by
@@ -354,13 +455,16 @@ theorem adjoin_le_subfield {K : Subfield E} (HF : Set.range (algebraMap F E) ⊆
rw [Set.union_subset_iff]
exact ⟨HF, HS⟩
#align intermediate_field.adjoin_le_subfield IntermediateField.adjoin_le_subfield
+-/
+#print IntermediateField.adjoin_subset_adjoin_iff /-
theorem adjoin_subset_adjoin_iff {F' : Type _} [Field F'] [Algebra F' E] {S S' : Set E} :
(adjoin F S : Set E) ⊆ adjoin F' S' ↔
Set.range (algebraMap F E) ⊆ adjoin F' S' ∧ S ⊆ adjoin F' S' :=
⟨fun h => ⟨trans (adjoin.range_algebraMap_subset _ _) h, trans (subset_adjoin _ _) h⟩,
fun ⟨hF, hS⟩ => Subfield.closure_le.mpr (Set.union_subset hF hS)⟩
#align intermediate_field.adjoin_subset_adjoin_iff IntermediateField.adjoin_subset_adjoin_iff
+-/
#print IntermediateField.adjoin_adjoin_left /-
/-- `F[S][T] = F[S ∪ T]` -/
@@ -398,6 +502,7 @@ theorem adjoin_adjoin_comm (T : Set E) :
#align intermediate_field.adjoin_adjoin_comm IntermediateField.adjoin_adjoin_comm
-/
+#print IntermediateField.adjoin_map /-
theorem adjoin_map {E' : Type _} [Field E'] [Algebra F E'] (f : E →ₐ[F] E') :
(adjoin F S).map f = adjoin F (f '' S) := by
ext x
@@ -408,11 +513,15 @@ theorem adjoin_map {E' : Type _} [Field E'] [Algebra F E'] (f : E →ₐ[F] E')
f.comp_algebra_map]
rfl
#align intermediate_field.adjoin_map IntermediateField.adjoin_map
+-/
+#print IntermediateField.algebra_adjoin_le_adjoin /-
theorem algebra_adjoin_le_adjoin : Algebra.adjoin F S ≤ (adjoin F S).toSubalgebra :=
Algebra.adjoin_le (subset_adjoin _ _)
#align intermediate_field.algebra_adjoin_le_adjoin IntermediateField.algebra_adjoin_le_adjoin
+-/
+#print IntermediateField.adjoin_eq_algebra_adjoin /-
theorem adjoin_eq_algebra_adjoin (inv_mem : ∀ x ∈ Algebra.adjoin F S, x⁻¹ ∈ Algebra.adjoin F S) :
(adjoin F S).toSubalgebra = Algebra.adjoin F S :=
le_antisymm
@@ -426,7 +535,9 @@ theorem adjoin_eq_algebra_adjoin (inv_mem : ∀ x ∈ Algebra.adjoin F S, x⁻¹
from adjoin_le_iff.mpr Algebra.subset_adjoin)
(algebra_adjoin_le_adjoin _ _)
#align intermediate_field.adjoin_eq_algebra_adjoin IntermediateField.adjoin_eq_algebra_adjoin
+-/
+#print IntermediateField.eq_adjoin_of_eq_algebra_adjoin /-
theorem eq_adjoin_of_eq_algebra_adjoin (K : IntermediateField F E)
(h : K.toSubalgebra = Algebra.adjoin F S) : K = adjoin F S :=
by
@@ -438,7 +549,9 @@ theorem eq_adjoin_of_eq_algebra_adjoin (K : IntermediateField F E)
rw [← h]
rfl
#align intermediate_field.eq_adjoin_of_eq_algebra_adjoin IntermediateField.eq_adjoin_of_eq_algebra_adjoin
+-/
+#print IntermediateField.adjoin_induction /-
@[elab_as_elim]
theorem adjoin_induction {s : Set E} {p : E → Prop} {x} (h : x ∈ adjoin F s) (Hs : ∀ x ∈ s, p x)
(Hmap : ∀ x, p (algebraMap F E x)) (Hadd : ∀ x y, p x → p y → p (x + y))
@@ -447,6 +560,7 @@ theorem adjoin_induction {s : Set E} {p : E → Prop} {x} (h : x ∈ adjoin F s)
Subfield.closure_induction h (fun x hx => Or.cases_on hx (fun ⟨x, hx⟩ => hx ▸ Hmap x) (Hs x))
((algebraMap F E).map_one ▸ Hmap 1) Hadd Hneg Hinv Hmul
#align intermediate_field.adjoin_induction IntermediateField.adjoin_induction
+-/
#print IntermediateField.Insert /-
--this definition of notation is courtesy of Kyle Miller on zulip
@@ -470,7 +584,6 @@ instance (priority := 900) insertNonempty {α : Type _} (s : Set α) : Insert s
#align intermediate_field.insert_nonempty IntermediateField.insertNonempty
-/
--- mathport name: «expr ⟮ ,⟯»
notation3:max K"⟮"(l ", "* => foldr (h t => Insert.insert t h) ∅)"⟯" => adjoin K l
section AdjoinSimple
@@ -478,23 +591,30 @@ section AdjoinSimple
variable (α : E)
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.mem_adjoin_simple_self /-
theorem mem_adjoin_simple_self : α ∈ F⟮⟯ :=
subset_adjoin F {α} (Set.mem_singleton α)
#align intermediate_field.mem_adjoin_simple_self IntermediateField.mem_adjoin_simple_self
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.AdjoinSimple.gen /-
/-- generator of `F⟮α⟯` -/
def AdjoinSimple.gen : F⟮⟯ :=
⟨α, mem_adjoin_simple_self F α⟩
#align intermediate_field.adjoin_simple.gen IntermediateField.AdjoinSimple.gen
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.AdjoinSimple.algebraMap_gen /-
@[simp]
theorem AdjoinSimple.algebraMap_gen : algebraMap F⟮⟯ E (AdjoinSimple.gen F α) = α :=
rfl
#align intermediate_field.adjoin_simple.algebra_map_gen IntermediateField.AdjoinSimple.algebraMap_gen
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.AdjoinSimple.isIntegral_gen /-
@[simp]
theorem AdjoinSimple.isIntegral_gen : IsIntegral F (AdjoinSimple.gen F α) ↔ IsIntegral F α :=
by
@@ -502,6 +622,7 @@ theorem AdjoinSimple.isIntegral_gen : IsIntegral F (AdjoinSimple.gen F α) ↔ I
rw [isIntegral_algebraMap_iff (algebraMap F⟮⟯ E).Injective]
infer_instance
#align intermediate_field.adjoin_simple.is_integral_gen IntermediateField.AdjoinSimple.isIntegral_gen
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
@@ -537,6 +658,7 @@ theorem adjoin_algebraic_toSubalgebra {S : Set E} (hS : ∀ x ∈ S, IsAlgebraic
-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.adjoin_simple_toSubalgebra_of_integral /-
theorem adjoin_simple_toSubalgebra_of_integral (hα : IsIntegral F α) :
F⟮⟯.toSubalgebra = Algebra.adjoin F {α} :=
by
@@ -544,7 +666,9 @@ theorem adjoin_simple_toSubalgebra_of_integral (hα : IsIntegral F α) :
rintro x (rfl : x = α)
rwa [isAlgebraic_iff_isIntegral]
#align intermediate_field.adjoin_simple_to_subalgebra_of_integral IntermediateField.adjoin_simple_toSubalgebra_of_integral
+-/
+#print IntermediateField.isSplittingField_iff /-
theorem isSplittingField_iff {p : F[X]} {K : IntermediateField F E} :
p.IsSplittingField F K ↔ p.Splits (algebraMap F K) ∧ K = adjoin F (p.rootSet E) :=
by
@@ -554,14 +678,18 @@ theorem isSplittingField_iff {p : F[X]} {K : IntermediateField F E} :
rw [← K.range_val, adjoin_algebraic_to_subalgebra fun x => isAlgebraic_of_mem_rootSet]
exact fun hp => (adjoin_root_set_eq_range hp K.val).symm.trans eq_comm
#align intermediate_field.is_splitting_field_iff IntermediateField.isSplittingField_iff
+-/
+#print IntermediateField.adjoin_rootSet_isSplittingField /-
theorem adjoin_rootSet_isSplittingField {p : F[X]} (hp : p.Splits (algebraMap F E)) :
p.IsSplittingField F (adjoin F (p.rootSet E)) :=
isSplittingField_iff.mpr ⟨splits_of_splits hp fun x hx => subset_adjoin F (p.rootSet E) hx, rfl⟩
#align intermediate_field.adjoin_root_set_is_splitting_field IntermediateField.adjoin_rootSet_isSplittingField
+-/
open scoped BigOperators
+#print IntermediateField.isSplittingField_iSup /-
/-- A compositum of splitting fields is a splitting field -/
theorem isSplittingField_iSup {ι : Type _} {t : ι → IntermediateField F E} {p : ι → F[X]}
{s : Finset ι} (h0 : ∏ i in s, p i ≠ 0) (h : ∀ i ∈ s, (p i).IsSplittingField F (t i)) :
@@ -578,16 +706,20 @@ theorem isSplittingField_iSup {ι : Type _} {t : ι → IntermediateField F E} {
simp only [root_set_prod p s h0, ← Set.iSup_eq_iUnion, (@gc F _ E _ _).l_iSup₂]
exact iSup_congr fun i => iSup_congr fun hi => (h i hi).2
#align intermediate_field.is_splitting_field_supr IntermediateField.isSplittingField_iSup
+-/
open Set CompleteLattice
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.adjoin_simple_le_iff /-
@[simp]
theorem adjoin_simple_le_iff {K : IntermediateField F E} : F⟮⟯ ≤ K ↔ α ∈ K :=
adjoin_le_iff.trans singleton_subset_iff
#align intermediate_field.adjoin_simple_le_iff IntermediateField.adjoin_simple_le_iff
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.adjoin_simple_isCompactElement /-
/-- Adjoining a single element is compact in the lattice of intermediate fields. -/
theorem adjoin_simple_isCompactElement (x : E) : IsCompactElement F⟮⟯ :=
by
@@ -618,8 +750,10 @@ theorem adjoin_simple_isCompactElement (x : E) : IsCompactElement F⟮⟯ :=
obtain ⟨-, ⟨E, rfl⟩, -, ⟨hE, rfl⟩, hx⟩ := key hx
exact ⟨E, hE, hx⟩
#align intermediate_field.adjoin_simple_is_compact_element IntermediateField.adjoin_simple_isCompactElement
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.adjoin_finset_isCompactElement /-
/-- Adjoining a finite subset is compact in the lattice of intermediate fields. -/
theorem adjoin_finset_isCompactElement (S : Finset E) :
IsCompactElement (adjoin F S : IntermediateField F E) :=
@@ -632,11 +766,14 @@ theorem adjoin_finset_isCompactElement (S : Finset E) :
rw [key, ← Finset.sup_eq_iSup]
exact finset_sup_compact_of_compact S fun x hx => adjoin_simple_is_compact_element x
#align intermediate_field.adjoin_finset_is_compact_element IntermediateField.adjoin_finset_isCompactElement
+-/
+#print IntermediateField.adjoin_finite_isCompactElement /-
/-- Adjoining a finite subset is compact in the lattice of intermediate fields. -/
theorem adjoin_finite_isCompactElement {S : Set E} (h : S.Finite) : IsCompactElement (adjoin F S) :=
Finite.coe_toFinset h ▸ adjoin_finset_isCompactElement h.toFinset
#align intermediate_field.adjoin_finite_is_compact_element IntermediateField.adjoin_finite_isCompactElement
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
/-- The lattice of intermediate fields is compactly generated. -/
@@ -648,6 +785,7 @@ instance : IsCompactlyGenerated (IntermediateField F E) :=
(le_antisymm (iSup_le fun i => iSup_le fun hi => adjoin_simple_le_iff.mpr hi) fun x hx =>
adjoin_simple_le_iff.mp (le_iSup_of_le x (le_iSup_of_le hx le_rfl)))⟩⟩⟩
+#print IntermediateField.exists_finset_of_mem_iSup /-
theorem exists_finset_of_mem_iSup {ι : Type _} {f : ι → IntermediateField F E} {x : E}
(hx : x ∈ ⨆ i, f i) : ∃ s : Finset ι, x ∈ ⨆ i ∈ s, f i :=
by
@@ -655,8 +793,10 @@ theorem exists_finset_of_mem_iSup {ι : Type _} {f : ι → IntermediateField F
simp only [adjoin_simple_le_iff] at this
exact this hx
#align intermediate_field.exists_finset_of_mem_supr IntermediateField.exists_finset_of_mem_iSup
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.exists_finset_of_mem_supr' /-
theorem exists_finset_of_mem_supr' {ι : Type _} {f : ι → IntermediateField F E} {x : E}
(hx : x ∈ ⨆ i, f i) : ∃ s : Finset (Σ i, f i), x ∈ ⨆ i ∈ s, F⟮⟯ :=
exists_finset_of_mem_iSup
@@ -665,7 +805,9 @@ theorem exists_finset_of_mem_supr' {ι : Type _} {f : ι → IntermediateField F
SetLike.le_def.mp (le_iSup_of_le ⟨i, x, h⟩ le_rfl) (mem_adjoin_simple_self F x))
hx)
#align intermediate_field.exists_finset_of_mem_supr' IntermediateField.exists_finset_of_mem_supr'
+-/
+#print IntermediateField.exists_finset_of_mem_supr'' /-
theorem exists_finset_of_mem_supr'' {ι : Type _} {f : ι → IntermediateField F E}
(h : ∀ i, Algebra.IsAlgebraic F (f i)) {x : E} (hx : x ∈ ⨆ i, f i) :
∃ s : Finset (Σ i, f i), x ∈ ⨆ i ∈ s, adjoin F ((minpoly F (i.2 : _)).rootSet E) :=
@@ -679,6 +821,7 @@ theorem exists_finset_of_mem_supr'' {ι : Type _} {f : ι → IntermediateField
rw [IntermediateField.minpoly_eq, Subtype.coe_mk, mem_root_set_of_ne, minpoly.aeval]
exact minpoly.ne_zero (is_integral_iff.mp (is_algebraic_iff_is_integral.mp (h i ⟨x, hx⟩)))
#align intermediate_field.exists_finset_of_mem_supr'' IntermediateField.exists_finset_of_mem_supr''
+-/
end AdjoinSimple
@@ -688,16 +831,20 @@ section AdjoinIntermediateFieldLattice
variable {F : Type _} [Field F] {E : Type _} [Field E] [Algebra F E] {α : E} {S : Set E}
+#print IntermediateField.adjoin_eq_bot_iff /-
@[simp]
theorem adjoin_eq_bot_iff : adjoin F S = ⊥ ↔ S ⊆ (⊥ : IntermediateField F E) := by
rw [eq_bot_iff, adjoin_le_iff]; rfl
#align intermediate_field.adjoin_eq_bot_iff IntermediateField.adjoin_eq_bot_iff
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.adjoin_simple_eq_bot_iff /-
@[simp]
theorem adjoin_simple_eq_bot_iff : F⟮⟯ = ⊥ ↔ α ∈ (⊥ : IntermediateField F E) := by
rw [adjoin_eq_bot_iff]; exact Set.singleton_subset_iff
#align intermediate_field.adjoin_simple_eq_bot_iff IntermediateField.adjoin_simple_eq_bot_iff
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
#print IntermediateField.adjoin_zero /-
@@ -737,45 +884,62 @@ open FiniteDimensional Module
variable {K L : IntermediateField F E}
+#print IntermediateField.rank_eq_one_iff /-
@[simp]
theorem rank_eq_one_iff : Module.rank F K = 1 ↔ K = ⊥ := by
rw [← to_subalgebra_eq_iff, ← rank_eq_rank_subalgebra, Subalgebra.rank_eq_one_iff,
bot_to_subalgebra]
#align intermediate_field.rank_eq_one_iff IntermediateField.rank_eq_one_iff
+-/
+#print IntermediateField.finrank_eq_one_iff /-
@[simp]
theorem finrank_eq_one_iff : finrank F K = 1 ↔ K = ⊥ := by
rw [← to_subalgebra_eq_iff, ← finrank_eq_finrank_subalgebra, Subalgebra.finrank_eq_one_iff,
bot_to_subalgebra]
#align intermediate_field.finrank_eq_one_iff IntermediateField.finrank_eq_one_iff
+-/
+#print IntermediateField.rank_bot /-
@[simp]
theorem rank_bot : Module.rank F (⊥ : IntermediateField F E) = 1 := by rw [rank_eq_one_iff]
#align intermediate_field.rank_bot IntermediateField.rank_bot
+-/
+#print IntermediateField.finrank_bot /-
@[simp]
theorem finrank_bot : finrank F (⊥ : IntermediateField F E) = 1 := by rw [finrank_eq_one_iff]
#align intermediate_field.finrank_bot IntermediateField.finrank_bot
+-/
+#print IntermediateField.rank_adjoin_eq_one_iff /-
theorem rank_adjoin_eq_one_iff : Module.rank F (adjoin F S) = 1 ↔ S ⊆ (⊥ : IntermediateField F E) :=
Iff.trans rank_eq_one_iff adjoin_eq_bot_iff
#align intermediate_field.rank_adjoin_eq_one_iff IntermediateField.rank_adjoin_eq_one_iff
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.rank_adjoin_simple_eq_one_iff /-
theorem rank_adjoin_simple_eq_one_iff : Module.rank F F⟮⟯ = 1 ↔ α ∈ (⊥ : IntermediateField F E) :=
by rw [rank_adjoin_eq_one_iff]; exact Set.singleton_subset_iff
#align intermediate_field.rank_adjoin_simple_eq_one_iff IntermediateField.rank_adjoin_simple_eq_one_iff
+-/
+#print IntermediateField.finrank_adjoin_eq_one_iff /-
theorem finrank_adjoin_eq_one_iff : finrank F (adjoin F S) = 1 ↔ S ⊆ (⊥ : IntermediateField F E) :=
Iff.trans finrank_eq_one_iff adjoin_eq_bot_iff
#align intermediate_field.finrank_adjoin_eq_one_iff IntermediateField.finrank_adjoin_eq_one_iff
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.finrank_adjoin_simple_eq_one_iff /-
theorem finrank_adjoin_simple_eq_one_iff : finrank F F⟮⟯ = 1 ↔ α ∈ (⊥ : IntermediateField F E) := by
rw [finrank_adjoin_eq_one_iff]; exact Set.singleton_subset_iff
#align intermediate_field.finrank_adjoin_simple_eq_one_iff IntermediateField.finrank_adjoin_simple_eq_one_iff
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.bot_eq_top_of_rank_adjoin_eq_one /-
/-- If `F⟮x⟯` has dimension `1` over `F` for every `x ∈ E` then `F = E`. -/
theorem bot_eq_top_of_rank_adjoin_eq_one (h : ∀ x : E, Module.rank F F⟮⟯ = 1) :
(⊥ : IntermediateField F E) = ⊤ := by
@@ -783,29 +947,37 @@ theorem bot_eq_top_of_rank_adjoin_eq_one (h : ∀ x : E, Module.rank F F⟮⟯ =
rw [iff_true_right IntermediateField.mem_top]
exact rank_adjoin_simple_eq_one_iff.mp (h x)
#align intermediate_field.bot_eq_top_of_rank_adjoin_eq_one IntermediateField.bot_eq_top_of_rank_adjoin_eq_one
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.bot_eq_top_of_finrank_adjoin_eq_one /-
theorem bot_eq_top_of_finrank_adjoin_eq_one (h : ∀ x : E, finrank F F⟮⟯ = 1) :
(⊥ : IntermediateField F E) = ⊤ := by
ext
rw [iff_true_right IntermediateField.mem_top]
exact finrank_adjoin_simple_eq_one_iff.mp (h x)
#align intermediate_field.bot_eq_top_of_finrank_adjoin_eq_one IntermediateField.bot_eq_top_of_finrank_adjoin_eq_one
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.subsingleton_of_rank_adjoin_eq_one /-
theorem subsingleton_of_rank_adjoin_eq_one (h : ∀ x : E, Module.rank F F⟮⟯ = 1) :
Subsingleton (IntermediateField F E) :=
subsingleton_of_bot_eq_top (bot_eq_top_of_rank_adjoin_eq_one h)
#align intermediate_field.subsingleton_of_rank_adjoin_eq_one IntermediateField.subsingleton_of_rank_adjoin_eq_one
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.subsingleton_of_finrank_adjoin_eq_one /-
theorem subsingleton_of_finrank_adjoin_eq_one (h : ∀ x : E, finrank F F⟮⟯ = 1) :
Subsingleton (IntermediateField F E) :=
subsingleton_of_bot_eq_top (bot_eq_top_of_finrank_adjoin_eq_one h)
#align intermediate_field.subsingleton_of_finrank_adjoin_eq_one IntermediateField.subsingleton_of_finrank_adjoin_eq_one
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.bot_eq_top_of_finrank_adjoin_le_one /-
/-- If `F⟮x⟯` has dimension `≤1` over `F` for every `x ∈ E` then `F = E`. -/
theorem bot_eq_top_of_finrank_adjoin_le_one [FiniteDimensional F E]
(h : ∀ x : E, finrank F F⟮⟯ ≤ 1) : (⊥ : IntermediateField F E) = ⊤ :=
@@ -813,12 +985,15 @@ theorem bot_eq_top_of_finrank_adjoin_le_one [FiniteDimensional F E]
apply bot_eq_top_of_finrank_adjoin_eq_one
exact fun x => by linarith [h x, show 0 < finrank F F⟮⟯ from finrank_pos]
#align intermediate_field.bot_eq_top_of_finrank_adjoin_le_one IntermediateField.bot_eq_top_of_finrank_adjoin_le_one
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.subsingleton_of_finrank_adjoin_le_one /-
theorem subsingleton_of_finrank_adjoin_le_one [FiniteDimensional F E]
(h : ∀ x : E, finrank F F⟮⟯ ≤ 1) : Subsingleton (IntermediateField F E) :=
subsingleton_of_bot_eq_top (bot_eq_top_of_finrank_adjoin_le_one h)
#align intermediate_field.subsingleton_of_finrank_adjoin_le_one IntermediateField.subsingleton_of_finrank_adjoin_le_one
+-/
end AdjoinRank
@@ -831,6 +1006,7 @@ variable {F : Type _} [Field F] {E : Type _} [Field E] [Algebra F E] {α : E}
variable {K : Type _} [Field K] [Algebra F K]
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.minpoly_gen /-
theorem minpoly_gen {α : E} (h : IsIntegral F α) : minpoly F (AdjoinSimple.gen F α) = minpoly F α :=
by
rw [← adjoin_simple.algebra_map_gen F α] at h
@@ -839,9 +1015,11 @@ theorem minpoly_gen {α : E} (h : IsIntegral F α) : minpoly F (AdjoinSimple.gen
minpoly.eq_of_algebraMap_eq inj ((isIntegral_algebraMap_iff inj).mp h)
(adjoin_simple.algebra_map_gen _ _).symm
#align intermediate_field.minpoly_gen IntermediateField.minpoly_gen
+-/
variable (F)
+#print IntermediateField.aeval_gen_minpoly /-
theorem aeval_gen_minpoly (α : E) : aeval (AdjoinSimple.gen F α) (minpoly F α) = 0 :=
by
ext
@@ -849,10 +1027,12 @@ theorem aeval_gen_minpoly (α : E) : aeval (AdjoinSimple.gen F α) (minpoly F α
conv in aeval α => rw [← adjoin_simple.algebra_map_gen F α]
exact (aeval_algebra_map_apply E (adjoin_simple.gen F α) _).symm
#align intermediate_field.aeval_gen_minpoly IntermediateField.aeval_gen_minpoly
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.adjoinRootEquivAdjoin /-
/-- algebra isomorphism between `adjoin_root` and `F⟮α⟯` -/
noncomputable def adjoinRootEquivAdjoin (h : IsIntegral F α) : AdjoinRoot (minpoly F α) ≃ₐ[F] F⟮⟯ :=
AlgEquiv.ofBijective
@@ -874,25 +1054,31 @@ noncomputable def adjoinRootEquivAdjoin (h : IsIntegral F α) : AdjoinRoot (minp
⟨AdjoinRoot.root (minpoly F α), by rw [RingHom.comp_apply, AdjoinRoot.lift_root];
rfl⟩)))
#align intermediate_field.adjoin_root_equiv_adjoin IntermediateField.adjoinRootEquivAdjoin
+-/
+#print IntermediateField.adjoinRootEquivAdjoin_apply_root /-
theorem adjoinRootEquivAdjoin_apply_root (h : IsIntegral F α) :
adjoinRootEquivAdjoin F h (AdjoinRoot.root (minpoly F α)) = AdjoinSimple.gen F α :=
AdjoinRoot.lift_root (aeval_gen_minpoly F α)
#align intermediate_field.adjoin_root_equiv_adjoin_apply_root IntermediateField.adjoinRootEquivAdjoin_apply_root
+-/
section PowerBasis
variable {L : Type _} [Field L] [Algebra K L]
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.powerBasisAux /-
/-- The elements `1, x, ..., x ^ (d - 1)` form a basis for `K⟮x⟯`,
where `d` is the degree of the minimal polynomial of `x`. -/
noncomputable def powerBasisAux {x : L} (hx : IsIntegral K x) :
Basis (Fin (minpoly K x).natDegree) K K⟮⟯ :=
(AdjoinRoot.powerBasis (minpoly.ne_zero hx)).Basis.map (adjoinRootEquivAdjoin K hx).toLinearEquiv
#align intermediate_field.power_basis_aux IntermediateField.powerBasisAux
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.adjoin.powerBasis /-
/-- The power basis `1, x, ..., x ^ (d - 1)` for `K⟮x⟯`,
where `d` is the degree of the minimal polynomial of `x`. -/
@[simps]
@@ -905,34 +1091,44 @@ noncomputable def adjoin.powerBasis {x : L} (hx : IsIntegral K x) : PowerBasis K
rw [power_basis_aux, Basis.map_apply, PowerBasis.basis_eq_pow, AlgEquiv.toLinearEquiv_apply,
AlgEquiv.map_pow, AdjoinRoot.powerBasis_gen, adjoin_root_equiv_adjoin_apply_root]
#align intermediate_field.adjoin.power_basis IntermediateField.adjoin.powerBasis
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.adjoin.finiteDimensional /-
theorem adjoin.finiteDimensional {x : L} (hx : IsIntegral K x) : FiniteDimensional K K⟮⟯ :=
PowerBasis.finiteDimensional (adjoin.powerBasis hx)
#align intermediate_field.adjoin.finite_dimensional IntermediateField.adjoin.finiteDimensional
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.adjoin.finrank /-
theorem adjoin.finrank {x : L} (hx : IsIntegral K x) :
FiniteDimensional.finrank K K⟮⟯ = (minpoly K x).natDegree :=
by
rw [PowerBasis.finrank (adjoin.power_basis hx : _)]
rfl
#align intermediate_field.adjoin.finrank IntermediateField.adjoin.finrank
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.minpoly.natDegree_le /-
theorem IntermediateField.minpoly.natDegree_le {x : L} [FiniteDimensional K L]
(hx : IsIntegral K x) : (minpoly K x).natDegree ≤ finrank K L :=
le_of_eq_of_le (IntermediateField.adjoin.finrank hx).symm K⟮⟯.toSubmodule.finrank_le
#align minpoly.nat_degree_le IntermediateField.minpoly.natDegree_le
+-/
+#print IntermediateField.minpoly.degree_le /-
theorem IntermediateField.minpoly.degree_le {x : L} [FiniteDimensional K L] (hx : IsIntegral K x) :
(minpoly K x).degree ≤ finrank K L :=
degree_le_of_natDegree_le (IntermediateField.minpoly.natDegree_le hx)
#align minpoly.degree_le IntermediateField.minpoly.degree_le
+-/
end PowerBasis
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.algHomAdjoinIntegralEquiv /-
/-- Algebra homomorphism `F⟮α⟯ →ₐ[F] K` are in bijection with the set of roots
of `minpoly α` in `K`. -/
noncomputable def algHomAdjoinIntegralEquiv (h : IsIntegral F α) :
@@ -941,20 +1137,25 @@ noncomputable def algHomAdjoinIntegralEquiv (h : IsIntegral F α) :
((Equiv.refl _).subtypeEquiv fun x => by
rw [adjoin.power_basis_gen, minpoly_gen h, Equiv.refl_apply])
#align intermediate_field.alg_hom_adjoin_integral_equiv IntermediateField.algHomAdjoinIntegralEquiv
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.fintypeOfAlgHomAdjoinIntegral /-
/-- Fintype of algebra homomorphism `F⟮α⟯ →ₐ[F] K` -/
noncomputable def fintypeOfAlgHomAdjoinIntegral (h : IsIntegral F α) : Fintype (F⟮⟯ →ₐ[F] K) :=
PowerBasis.AlgHom.fintype (adjoin.powerBasis h)
#align intermediate_field.fintype_of_alg_hom_adjoin_integral IntermediateField.fintypeOfAlgHomAdjoinIntegral
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.card_algHom_adjoin_integral /-
theorem card_algHom_adjoin_integral (h : IsIntegral F α) (h_sep : (minpoly F α).Separable)
(h_splits : (minpoly F α).Splits (algebraMap F K)) :
@Fintype.card (F⟮⟯ →ₐ[F] K) (fintypeOfAlgHomAdjoinIntegral F h) = (minpoly F α).natDegree := by
rw [AlgHom.card_of_powerBasis] <;>
simp only [adjoin.power_basis_dim, adjoin.power_basis_gen, minpoly_gen h, h_sep, h_splits]
#align intermediate_field.card_alg_hom_adjoin_integral IntermediateField.card_algHom_adjoin_integral
+-/
end AdjoinIntegralElement
@@ -976,25 +1177,34 @@ theorem fg_adjoin_finset (t : Finset E) : (adjoin F (↑t : Set E)).FG :=
#align intermediate_field.fg_adjoin_finset IntermediateField.fg_adjoin_finset
-/
+#print IntermediateField.fg_def /-
theorem fg_def {S : IntermediateField F E} : S.FG ↔ ∃ t : Set E, Set.Finite t ∧ adjoin F t = S :=
Iff.symm Set.exists_finite_iff_finset
#align intermediate_field.fg_def IntermediateField.fg_def
+-/
+#print IntermediateField.fg_bot /-
theorem fg_bot : (⊥ : IntermediateField F E).FG :=
⟨∅, adjoin_empty F E⟩
#align intermediate_field.fg_bot IntermediateField.fg_bot
+-/
+#print IntermediateField.fG_of_fG_toSubalgebra /-
theorem fG_of_fG_toSubalgebra (S : IntermediateField F E) (h : S.toSubalgebra.FG) : S.FG :=
by
cases' h with t ht
exact ⟨t, (eq_adjoin_of_eq_algebra_adjoin _ _ _ ht.symm).symm⟩
#align intermediate_field.fg_of_fg_to_subalgebra IntermediateField.fG_of_fG_toSubalgebra
+-/
+#print IntermediateField.fg_of_noetherian /-
theorem fg_of_noetherian (S : IntermediateField F E) [IsNoetherian F E] : S.FG :=
S.fG_of_fG_toSubalgebra S.toSubalgebra.fg_of_noetherian
#align intermediate_field.fg_of_noetherian IntermediateField.fg_of_noetherian
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.induction_on_adjoin_finset /-
theorem induction_on_adjoin_finset (S : Finset E) (P : IntermediateField F E → Prop) (base : P ⊥)
(ih : ∀ (K : IntermediateField F E), ∀ x ∈ S, P K → P (K⟮⟯.restrictScalars F)) :
P (adjoin F ↑S) := by
@@ -1004,8 +1214,10 @@ theorem induction_on_adjoin_finset (S : Finset E) (P : IntermediateField F E →
rw [Finset.coe_insert, Set.insert_eq, Set.union_comm, ← adjoin_adjoin_left]
exact ih (adjoin F s) a h1 h4
#align intermediate_field.induction_on_adjoin_finset IntermediateField.induction_on_adjoin_finset
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.induction_on_adjoin_fg /-
theorem induction_on_adjoin_fg (P : IntermediateField F E → Prop) (base : P ⊥)
(ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮⟯.restrictScalars F))
(K : IntermediateField F E) (hK : K.FG) : P K :=
@@ -1013,14 +1225,17 @@ theorem induction_on_adjoin_fg (P : IntermediateField F E → Prop) (base : P
obtain ⟨S, rfl⟩ := hK
exact induction_on_adjoin_finset S P base fun K x _ hK => ih K x hK
#align intermediate_field.induction_on_adjoin_fg IntermediateField.induction_on_adjoin_fg
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.induction_on_adjoin /-
theorem induction_on_adjoin [fd : FiniteDimensional F E] (P : IntermediateField F E → Prop)
(base : P ⊥) (ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮⟯.restrictScalars F))
(K : IntermediateField F E) : P K :=
letI : IsNoetherian F E := IsNoetherian.iff_fg.2 inferInstance
induction_on_adjoin_fg P base ih K K.fg_of_noetherian
#align intermediate_field.induction_on_adjoin IntermediateField.induction_on_adjoin
+-/
end Induction
@@ -1063,10 +1278,13 @@ noncomputable instance : OrderBot (Lifts F E K)
noncomputable instance : Inhabited (Lifts F E K) :=
⟨⊥⟩
+#print IntermediateField.Lifts.eq_of_le /-
theorem Lifts.eq_of_le {x y : Lifts F E K} (hxy : x ≤ y) (s : x.1) : x.2 s = y.2 ⟨s, hxy.1 s.Mem⟩ :=
hxy.2 s ⟨s, hxy.1 s.Mem⟩ rfl
#align intermediate_field.lifts.eq_of_le IntermediateField.Lifts.eq_of_le
+-/
+#print IntermediateField.Lifts.exists_max_two /-
theorem Lifts.exists_max_two {c : Set (Lifts F E K)} {x y : Lifts F E K} (hc : IsChain (· ≤ ·) c)
(hx : x ∈ Insert.insert ⊥ c) (hy : y ∈ Insert.insert ⊥ c) :
∃ z : Lifts F E K, z ∈ Insert.insert ⊥ c ∧ x ≤ z ∧ y ≤ z :=
@@ -1075,7 +1293,9 @@ theorem Lifts.exists_max_two {c : Set (Lifts F E K)} {x y : Lifts F E K} (hc : I
· exact ⟨y, hy, hxy, le_refl y⟩
· exact ⟨x, hx, le_refl x, hyx⟩
#align intermediate_field.lifts.exists_max_two IntermediateField.Lifts.exists_max_two
+-/
+#print IntermediateField.Lifts.exists_max_three /-
theorem Lifts.exists_max_three {c : Set (Lifts F E K)} {x y z : Lifts F E K}
(hc : IsChain (· ≤ ·) c) (hx : x ∈ Insert.insert ⊥ c) (hy : y ∈ Insert.insert ⊥ c)
(hz : z ∈ Insert.insert ⊥ c) :
@@ -1085,7 +1305,9 @@ theorem Lifts.exists_max_three {c : Set (Lifts F E K)} {x y z : Lifts F E K}
obtain ⟨w, hw, hzw, hvw⟩ := lifts.exists_max_two hc hz hv
exact ⟨w, hw, le_trans hxv hvw, le_trans hyv hvw, hzw⟩
#align intermediate_field.lifts.exists_max_three IntermediateField.Lifts.exists_max_three
+-/
+#print IntermediateField.Lifts.upperBoundIntermediateField /-
/-- An upper bound on a chain of lifts -/
def Lifts.upperBoundIntermediateField {c : Set (Lifts F E K)} (hc : IsChain (· ≤ ·) c) :
IntermediateField F E
@@ -1105,7 +1327,9 @@ def Lifts.upperBoundIntermediateField {c : Set (Lifts F E K)} (hc : IsChain (·
exact ⟨z, hz, z.1.mul_mem (hxz.1 ha) (hyz.1 hb)⟩
algebraMap_mem' s := ⟨⊥, Set.mem_insert ⊥ c, algebraMap_mem ⊥ s⟩
#align intermediate_field.lifts.upper_bound_intermediate_field IntermediateField.Lifts.upperBoundIntermediateField
+-/
+#print IntermediateField.Lifts.upperBoundAlgHom /-
/-- The lift on the upper bound on a chain of lifts -/
noncomputable def Lifts.upperBoundAlgHom {c : Set (Lifts F E K)} (hc : IsChain (· ≤ ·) c) :
Lifts.upperBoundIntermediateField hc →ₐ[F] K
@@ -1129,12 +1353,16 @@ noncomputable def Lifts.upperBoundAlgHom {c : Set (Lifts F E K)} (hc : IsChain (
rfl
commutes' _ := AlgHom.commutes _ _
#align intermediate_field.lifts.upper_bound_alg_hom IntermediateField.Lifts.upperBoundAlgHom
+-/
+#print IntermediateField.Lifts.upperBound /-
/-- An upper bound on a chain of lifts -/
noncomputable def Lifts.upperBound {c : Set (Lifts F E K)} (hc : IsChain (· ≤ ·) c) : Lifts F E K :=
⟨Lifts.upperBoundIntermediateField hc, Lifts.upperBoundAlgHom hc⟩
#align intermediate_field.lifts.upper_bound IntermediateField.Lifts.upperBound
+-/
+#print IntermediateField.Lifts.exists_upper_bound /-
theorem Lifts.exists_upper_bound (c : Set (Lifts F E K)) (hc : IsChain (· ≤ ·) c) :
∃ ub, ∀ a ∈ c, a ≤ ub :=
⟨Lifts.upperBound hc, by
@@ -1148,6 +1376,7 @@ theorem Lifts.exists_upper_bound (c : Set (Lifts F E K)) (hc : IsChain (· ≤
rw [lifts.eq_of_le hxz, lifts.eq_of_le hyz]
exact congr_arg z.2 (Subtype.ext hst)⟩
#align intermediate_field.lifts.exists_upper_bound IntermediateField.Lifts.exists_upper_bound
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
@@ -1175,6 +1404,7 @@ noncomputable def Lifts.liftOfSplits (x : Lifts F E K) {s : E} (h1 : IsIntegral
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.Lifts.le_lifts_of_splits /-
theorem Lifts.le_lifts_of_splits (x : Lifts F E K) {s : E} (h1 : IsIntegral F s)
(h2 : (minpoly F s).Splits (algebraMap F K)) : x ≤ x.lift_of_splits h1 h2 :=
⟨fun z hz => algebraMap_mem x.1⟮⟯ ⟨z, hz⟩, fun t u htu =>
@@ -1184,17 +1414,23 @@ theorem Lifts.le_lifts_of_splits (x : Lifts F E K) {s : E} (h1 : IsIntegral F s)
letI : Algebra x.1 K := x.2.toRingHom.toAlgebra
exact AlgHom.commutes _ t)⟩
#align intermediate_field.lifts.le_lifts_of_splits IntermediateField.Lifts.le_lifts_of_splits
+-/
+#print IntermediateField.Lifts.mem_lifts_of_splits /-
theorem Lifts.mem_lifts_of_splits (x : Lifts F E K) {s : E} (h1 : IsIntegral F s)
(h2 : (minpoly F s).Splits (algebraMap F K)) : s ∈ (x.lift_of_splits h1 h2).1 :=
mem_adjoin_simple_self x.1 s
#align intermediate_field.lifts.mem_lifts_of_splits IntermediateField.Lifts.mem_lifts_of_splits
+-/
+#print IntermediateField.Lifts.exists_lift_of_splits /-
theorem Lifts.exists_lift_of_splits (x : Lifts F E K) {s : E} (h1 : IsIntegral F s)
(h2 : (minpoly F s).Splits (algebraMap F K)) : ∃ y, x ≤ y ∧ s ∈ y.1 :=
⟨x.lift_of_splits h1 h2, x.le_lifts_of_splits h1 h2, x.mem_lifts_of_splits h1 h2⟩
#align intermediate_field.lifts.exists_lift_of_splits IntermediateField.Lifts.exists_lift_of_splits
+-/
+#print IntermediateField.algHom_mk_adjoin_splits /-
theorem algHom_mk_adjoin_splits
(hK : ∀ s ∈ S, IsIntegral F (s : E) ∧ (minpoly F s).Splits (algebraMap F K)) :
Nonempty (adjoin F S →ₐ[F] K) :=
@@ -1207,7 +1443,9 @@ theorem algHom_mk_adjoin_splits
rcases x.exists_lift_of_splits (hK s hs).1 (hK s hs).2 with ⟨y, h1, h2⟩
rwa [hx y h1] at h2
#align intermediate_field.alg_hom_mk_adjoin_splits IntermediateField.algHom_mk_adjoin_splits
+-/
+#print IntermediateField.algHom_mk_adjoin_splits' /-
theorem algHom_mk_adjoin_splits' (hS : adjoin F S = ⊤)
(hK : ∀ x ∈ S, IsIntegral F (x : E) ∧ (minpoly F x).Splits (algebraMap F K)) :
Nonempty (E →ₐ[F] K) := by
@@ -1215,6 +1453,7 @@ theorem algHom_mk_adjoin_splits' (hS : adjoin F S = ⊤)
rw [hS] at ϕ
exact ⟨ϕ.comp top_equiv.symm.to_alg_hom⟩
#align intermediate_field.alg_hom_mk_adjoin_splits' IntermediateField.algHom_mk_adjoin_splits'
+-/
end AlgHomMkAdjoinSplits
@@ -1222,10 +1461,13 @@ section Supremum
variable {K L : Type _} [Field K] [Field L] [Algebra K L] (E1 E2 : IntermediateField K L)
+#print IntermediateField.le_sup_toSubalgebra /-
theorem le_sup_toSubalgebra : E1.toSubalgebra ⊔ E2.toSubalgebra ≤ (E1 ⊔ E2).toSubalgebra :=
sup_le (show E1 ≤ E1 ⊔ E2 from le_sup_left) (show E2 ≤ E1 ⊔ E2 from le_sup_right)
#align intermediate_field.le_sup_to_subalgebra IntermediateField.le_sup_toSubalgebra
+-/
+#print IntermediateField.sup_toSubalgebra /-
theorem sup_toSubalgebra [h1 : FiniteDimensional K E1] [h2 : FiniteDimensional K E2] :
(E1 ⊔ E2).toSubalgebra = E1.toSubalgebra ⊔ E2.toSubalgebra :=
by
@@ -1248,7 +1490,9 @@ theorem sup_toSubalgebra [h1 : FiniteDimensional K E1] [h2 : FiniteDimensional K
(is_integral_sup.mpr ⟨Algebra.isIntegral_of_finite K E1, Algebra.isIntegral_of_finite K E2⟩)
(Field.toIsField K)
#align intermediate_field.sup_to_subalgebra IntermediateField.sup_toSubalgebra
+-/
+#print IntermediateField.finiteDimensional_sup /-
instance finiteDimensional_sup [h1 : FiniteDimensional K E1] [h2 : FiniteDimensional K E2] :
FiniteDimensional K ↥(E1 ⊔ E2) :=
by
@@ -1259,7 +1503,9 @@ instance finiteDimensional_sup [h1 : FiniteDimensional K E1] [h2 : FiniteDimensi
rwa [this] at h
rw [Algebra.TensorProduct.productMap_range, E1.range_val, E2.range_val, sup_to_subalgebra]
#align intermediate_field.finite_dimensional_sup IntermediateField.finiteDimensional_sup
+-/
+#print IntermediateField.finiteDimensional_iSup_of_finite /-
instance finiteDimensional_iSup_of_finite {ι : Type _} {t : ι → IntermediateField K L}
[h : Finite ι] [∀ i, FiniteDimensional K (t i)] :
FiniteDimensional K (⨆ i, t i : IntermediateField K L) :=
@@ -1276,7 +1522,9 @@ instance finiteDimensional_iSup_of_finite {ι : Type _} {t : ι → Intermediate
rw [iSup_insert]
exact IntermediateField.finiteDimensional_sup _ _
#align intermediate_field.finite_dimensional_supr_of_finite IntermediateField.finiteDimensional_iSup_of_finite
+-/
+#print IntermediateField.finiteDimensional_iSup_of_finset /-
instance finiteDimensional_iSup_of_finset {ι : Type _} {f : ι → IntermediateField K L}
{s : Finset ι} [h : ∀ i ∈ s, FiniteDimensional K (f i)] :
FiniteDimensional K (⨆ i ∈ s, f i : IntermediateField K L) :=
@@ -1287,8 +1535,10 @@ instance finiteDimensional_iSup_of_finset {ι : Type _} {f : ι → Intermediate
(iSup_le fun i => le_iSup_of_le i (le_iSup_of_le i.2 le_rfl))
exact this.symm ▸ IntermediateField.finiteDimensional_iSup_of_finite
#align intermediate_field.finite_dimensional_supr_of_finset IntermediateField.finiteDimensional_iSup_of_finset
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.isAlgebraic_iSup /-
/-- A compositum of algebraic extensions is algebraic -/
theorem isAlgebraic_iSup {ι : Type _} {f : ι → IntermediateField K L}
(h : ∀ i, Algebra.IsAlgebraic K (f i)) :
@@ -1301,6 +1551,7 @@ theorem isAlgebraic_iSup {ι : Type _} {f : ι → IntermediateField K L}
adjoin.finite_dimensional (is_integral_iff.1 (isAlgebraic_iff_isIntegral.1 (h i x)))
apply Algebra.isAlgebraic_of_finite
#align intermediate_field.is_algebraic_supr IntermediateField.isAlgebraic_iSup
+-/
end Supremum
@@ -1316,6 +1567,7 @@ open IntermediateField
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print PowerBasis.equivAdjoinSimple /-
/-- `pb.equiv_adjoin_simple` is the equivalence between `K⟮pb.gen⟯` and `L` itself. -/
noncomputable def equivAdjoinSimple (pb : PowerBasis K L) : K⟮⟯ ≃ₐ[K] L :=
(adjoin.powerBasis pb.isIntegral_gen).equivOfMinpoly pb
@@ -1323,30 +1575,39 @@ noncomputable def equivAdjoinSimple (pb : PowerBasis K L) : K⟮⟯ ≃ₐ[K] L
(adjoin.powerBasis pb.isIntegral_gen).isIntegral_gen
(by rw [adjoin.power_basis_gen, adjoin_simple.algebra_map_gen]))
#align power_basis.equiv_adjoin_simple PowerBasis.equivAdjoinSimple
+-/
+#print PowerBasis.equivAdjoinSimple_aeval /-
@[simp]
theorem equivAdjoinSimple_aeval (pb : PowerBasis K L) (f : K[X]) :
pb.equivAdjoinSimple (aeval (AdjoinSimple.gen K pb.gen) f) = aeval pb.gen f :=
equivOfMinpoly_aeval _ pb _ f
#align power_basis.equiv_adjoin_simple_aeval PowerBasis.equivAdjoinSimple_aeval
+-/
+#print PowerBasis.equivAdjoinSimple_gen /-
@[simp]
theorem equivAdjoinSimple_gen (pb : PowerBasis K L) :
pb.equivAdjoinSimple (AdjoinSimple.gen K pb.gen) = pb.gen :=
equivOfMinpoly_gen _ pb _
#align power_basis.equiv_adjoin_simple_gen PowerBasis.equivAdjoinSimple_gen
+-/
+#print PowerBasis.equivAdjoinSimple_symm_aeval /-
@[simp]
theorem equivAdjoinSimple_symm_aeval (pb : PowerBasis K L) (f : K[X]) :
pb.equivAdjoinSimple.symm (aeval pb.gen f) = aeval (AdjoinSimple.gen K pb.gen) f := by
rw [equiv_adjoin_simple, equiv_of_minpoly_symm, equiv_of_minpoly_aeval, adjoin.power_basis_gen]
#align power_basis.equiv_adjoin_simple_symm_aeval PowerBasis.equivAdjoinSimple_symm_aeval
+-/
+#print PowerBasis.equivAdjoinSimple_symm_gen /-
@[simp]
theorem equivAdjoinSimple_symm_gen (pb : PowerBasis K L) :
pb.equivAdjoinSimple.symm pb.gen = AdjoinSimple.gen K pb.gen := by
rw [equiv_adjoin_simple, equiv_of_minpoly_symm, equiv_of_minpoly_gen, adjoin.power_basis_gen]
#align power_basis.equiv_adjoin_simple_symm_gen PowerBasis.equivAdjoinSimple_symm_gen
+-/
end PowerBasis
bump to nightly-2023-06-10-07
mathlib commit https://github.com/leanprover-community/mathlib/commit/a3e83f0fa4391c8740f7d773a7a9b74e311ae2a3
Diff
@@ -564,7 +564,7 @@ open scoped BigOperators
/-- A compositum of splitting fields is a splitting field -/
theorem isSplittingField_iSup {ι : Type _} {t : ι → IntermediateField F E} {p : ι → F[X]}
- {s : Finset ι} (h0 : (∏ i in s, p i) ≠ 0) (h : ∀ i ∈ s, (p i).IsSplittingField F (t i)) :
+ {s : Finset ι} (h0 : ∏ i in s, p i ≠ 0) (h : ∀ i ∈ s, (p i).IsSplittingField F (t i)) :
(∏ i in s, p i).IsSplittingField F (⨆ i ∈ s, t i : IntermediateField F E) :=
by
let K : IntermediateField F E := ⨆ i ∈ s, t i
bump to nightly-2023-06-10-03
mathlib commit https://github.com/leanprover-community/mathlib/commit/a3e83f0fa4391c8740f7d773a7a9b74e311ae2a3
Diff
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
! This file was ported from Lean 3 source module field_theory.adjoin
-! leanprover-community/mathlib commit df76f43357840485b9d04ed5dee5ab115d420e87
+! leanprover-community/mathlib commit 6b31d1eebd64eab86d5bd9936bfaada6ca8b5842
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
@@ -16,6 +16,9 @@ import Mathbin.RingTheory.TensorProduct
/-!
# Adjoining Elements to Fields
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
In this file we introduce the notion of adjoining elements to fields.
This isn't quite the same as adjoining elements to rings.
For example, `algebra.adjoin K {x}` might not include `x⁻¹`.
bump to nightly-2023-06-08-15
mathlib commit https://github.com/leanprover-community/mathlib/commit/31c24aa72e7b3e5ed97a8412470e904f82b81004
Diff
@@ -42,11 +42,13 @@ section AdjoinDef
variable (F : Type _) [Field F] {E : Type _} [Field E] [Algebra F E] (S : Set E)
+#print IntermediateField.adjoin /-
/-- `adjoin F S` extends a field `F` by adjoining a set `S ⊆ E`. -/
def adjoin : IntermediateField F E :=
{ Subfield.closure (Set.range (algebraMap F E) ∪ S) with
algebraMap_mem' := fun x => Subfield.subset_closure (Or.inl (Set.mem_range_self x)) }
#align intermediate_field.adjoin IntermediateField.adjoin
+-/
end AdjoinDef
@@ -260,26 +262,26 @@ theorem restrictScalars_top {K : Type _} [Field K] [Algebra K E] [Algebra K F]
rfl
#align intermediate_field.restrict_scalars_top IntermediateField.restrictScalars_top
-theorem AlgHom.fieldRange_eq_map {K : Type _} [Field K] [Algebra F K] (f : E →ₐ[F] K) :
- f.fieldRange = IntermediateField.map f ⊤ :=
+theorem IntermediateField.AlgHom.fieldRange_eq_map {K : Type _} [Field K] [Algebra F K]
+ (f : E →ₐ[F] K) : f.fieldRange = IntermediateField.map f ⊤ :=
SetLike.ext' Set.image_univ.symm
-#align alg_hom.field_range_eq_map AlgHom.fieldRange_eq_map
+#align alg_hom.field_range_eq_map IntermediateField.AlgHom.fieldRange_eq_map
-theorem AlgHom.map_fieldRange {K L : Type _} [Field K] [Field L] [Algebra F K] [Algebra F L]
- (f : E →ₐ[F] K) (g : K →ₐ[F] L) : f.fieldRange.map g = (g.comp f).fieldRange :=
+theorem IntermediateField.AlgHom.map_fieldRange {K L : Type _} [Field K] [Field L] [Algebra F K]
+ [Algebra F L] (f : E →ₐ[F] K) (g : K →ₐ[F] L) : f.fieldRange.map g = (g.comp f).fieldRange :=
SetLike.ext' (Set.range_comp g f).symm
-#align alg_hom.map_field_range AlgHom.map_fieldRange
+#align alg_hom.map_field_range IntermediateField.AlgHom.map_fieldRange
-theorem AlgHom.fieldRange_eq_top {K : Type _} [Field K] [Algebra F K] {f : E →ₐ[F] K} :
- f.fieldRange = ⊤ ↔ Function.Surjective f :=
+theorem IntermediateField.AlgHom.fieldRange_eq_top {K : Type _} [Field K] [Algebra F K]
+ {f : E →ₐ[F] K} : f.fieldRange = ⊤ ↔ Function.Surjective f :=
SetLike.ext'_iff.trans Set.range_iff_surjective
-#align alg_hom.field_range_eq_top AlgHom.fieldRange_eq_top
+#align alg_hom.field_range_eq_top IntermediateField.AlgHom.fieldRange_eq_top
@[simp]
-theorem AlgEquiv.fieldRange_eq_top {K : Type _} [Field K] [Algebra F K] (f : E ≃ₐ[F] K) :
- (f : E →ₐ[F] K).fieldRange = ⊤ :=
- AlgHom.fieldRange_eq_top.mpr f.Surjective
-#align alg_equiv.field_range_eq_top AlgEquiv.fieldRange_eq_top
+theorem IntermediateField.AlgEquiv.fieldRange_eq_top {K : Type _} [Field K] [Algebra F K]
+ (f : E ≃ₐ[F] K) : (f : E →ₐ[F] K).fieldRange = ⊤ :=
+ IntermediateField.AlgHom.fieldRange_eq_top.mpr f.Surjective
+#align alg_equiv.field_range_eq_top IntermediateField.AlgEquiv.fieldRange_eq_top
end Lattice
@@ -357,6 +359,7 @@ theorem adjoin_subset_adjoin_iff {F' : Type _} [Field F'] [Algebra F' E] {S S' :
fun ⟨hF, hS⟩ => Subfield.closure_le.mpr (Set.union_subset hF hS)⟩
#align intermediate_field.adjoin_subset_adjoin_iff IntermediateField.adjoin_subset_adjoin_iff
+#print IntermediateField.adjoin_adjoin_left /-
/-- `F[S][T] = F[S ∪ T]` -/
theorem adjoin_adjoin_left (T : Set E) :
(adjoin (adjoin F S) T).restrictScalars _ = adjoin F (S ∪ T) :=
@@ -369,7 +372,9 @@ theorem adjoin_adjoin_left (T : Set E) :
· exact subset_adjoin_of_subset_left _ (adjoin.range_algebra_map_subset _ _)
· exact Set.union_subset (subset_adjoin_of_subset_left _ (subset_adjoin _ _)) (subset_adjoin _ _)
#align intermediate_field.adjoin_adjoin_left IntermediateField.adjoin_adjoin_left
+-/
+#print IntermediateField.adjoin_insert_adjoin /-
@[simp]
theorem adjoin_insert_adjoin (x : E) :
adjoin F (insert x (adjoin F S : Set E)) = adjoin F (insert x S) :=
@@ -380,12 +385,15 @@ theorem adjoin_insert_adjoin (x : E) :
adjoin_le_iff.mpr (subset_adjoin_of_subset_right _ _ (Set.subset_insert _ _))⟩))
(adjoin.mono _ _ _ (Set.insert_subset_insert (subset_adjoin _ _)))
#align intermediate_field.adjoin_insert_adjoin IntermediateField.adjoin_insert_adjoin
+-/
+#print IntermediateField.adjoin_adjoin_comm /-
/-- `F[S][T] = F[T][S]` -/
theorem adjoin_adjoin_comm (T : Set E) :
(adjoin (adjoin F S) T).restrictScalars F = (adjoin (adjoin F T) S).restrictScalars F := by
rw [adjoin_adjoin_left, adjoin_adjoin_left, Set.union_comm]
#align intermediate_field.adjoin_adjoin_comm IntermediateField.adjoin_adjoin_comm
+-/
theorem adjoin_map {E' : Type _} [Field E'] [Algebra F E'] (f : E →ₐ[F] E') :
(adjoin F S).map f = adjoin F (f '' S) := by
@@ -437,6 +445,7 @@ theorem adjoin_induction {s : Set E} {p : E → Prop} {x} (h : x ∈ adjoin F s)
((algebraMap F E).map_one ▸ Hmap 1) Hadd Hneg Hinv Hmul
#align intermediate_field.adjoin_induction IntermediateField.adjoin_induction
+#print IntermediateField.Insert /-
--this definition of notation is courtesy of Kyle Miller on zulip
/-- Variation on `set.insert` to enable good notation for adjoining elements to fields.
Used to preferentially use `singleton` rather than `insert` when adjoining one element.
@@ -444,14 +453,19 @@ Used to preferentially use `singleton` rather than `insert` when adjoining one e
class Insert {α : Type _} (s : Set α) where
insert : α → Set α
#align intermediate_field.insert IntermediateField.Insert
+-/
+#print IntermediateField.insertEmpty /-
instance (priority := 1000) insertEmpty {α : Type _} : Insert (∅ : Set α)
where insert x := @singleton _ _ Set.hasSingleton x
#align intermediate_field.insert_empty IntermediateField.insertEmpty
+-/
+#print IntermediateField.insertNonempty /-
instance (priority := 900) insertNonempty {α : Type _} (s : Set α) : Insert s
where insert x := Insert.insert x s
#align intermediate_field.insert_nonempty IntermediateField.insertNonempty
+-/
-- mathport name: «expr ⟮ ,⟯»
notation3:max K"⟮"(l ", "* => foldr (h t => Insert.insert t h) ∅)"⟯" => adjoin K l
@@ -489,20 +503,25 @@ theorem AdjoinSimple.isIntegral_gen : IsIntegral F (AdjoinSimple.gen F α) ↔ I
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.adjoin_simple_adjoin_simple /-
theorem adjoin_simple_adjoin_simple (β : E) : F⟮⟯⟮⟯.restrictScalars F = F⟮⟯ :=
adjoin_adjoin_left _ _ _
#align intermediate_field.adjoin_simple_adjoin_simple IntermediateField.adjoin_simple_adjoin_simple
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.adjoin_simple_comm /-
theorem adjoin_simple_comm (β : E) : F⟮⟯⟮⟯.restrictScalars F = F⟮⟯⟮⟯.restrictScalars F :=
adjoin_adjoin_comm _ _ _
#align intermediate_field.adjoin_simple_comm IntermediateField.adjoin_simple_comm
+-/
variable {F} {α}
+#print IntermediateField.adjoin_algebraic_toSubalgebra /-
theorem adjoin_algebraic_toSubalgebra {S : Set E} (hS : ∀ x ∈ S, IsAlgebraic F x) :
(IntermediateField.adjoin F S).toSubalgebra = Algebra.adjoin F S :=
by
@@ -512,6 +531,7 @@ theorem adjoin_algebraic_toSubalgebra {S : Set E} (hS : ∀ x ∈ S, IsAlgebraic
have := isField_of_isIntegral_of_isField' this (Field.toIsField F)
rw [← ((Algebra.adjoin F S).toIntermediateField' this).eq_adjoin_of_eq_algebra_adjoin F S] <;> rfl
#align intermediate_field.adjoin_algebraic_to_subalgebra IntermediateField.adjoin_algebraic_toSubalgebra
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
theorem adjoin_simple_toSubalgebra_of_integral (hα : IsIntegral F α) :
@@ -677,28 +697,36 @@ theorem adjoin_simple_eq_bot_iff : F⟮⟯ = ⊥ ↔ α ∈ (⊥ : IntermediateF
#align intermediate_field.adjoin_simple_eq_bot_iff IntermediateField.adjoin_simple_eq_bot_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.adjoin_zero /-
@[simp]
theorem adjoin_zero : F⟮⟯ = ⊥ :=
adjoin_simple_eq_bot_iff.mpr (zero_mem ⊥)
#align intermediate_field.adjoin_zero IntermediateField.adjoin_zero
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.adjoin_one /-
@[simp]
theorem adjoin_one : F⟮⟯ = ⊥ :=
adjoin_simple_eq_bot_iff.mpr (one_mem ⊥)
#align intermediate_field.adjoin_one IntermediateField.adjoin_one
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.adjoin_int /-
@[simp]
theorem adjoin_int (n : ℤ) : F⟮⟯ = ⊥ :=
adjoin_simple_eq_bot_iff.mpr (coe_int_mem ⊥ n)
#align intermediate_field.adjoin_int IntermediateField.adjoin_int
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.adjoin_nat /-
@[simp]
theorem adjoin_nat (n : ℕ) : F⟮⟯ = ⊥ :=
adjoin_simple_eq_bot_iff.mpr (coe_nat_mem ⊥ n)
#align intermediate_field.adjoin_nat IntermediateField.adjoin_nat
+-/
section AdjoinRank
@@ -889,15 +917,15 @@ theorem adjoin.finrank {x : L} (hx : IsIntegral K x) :
#align intermediate_field.adjoin.finrank IntermediateField.adjoin.finrank
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
-theorem minpoly.natDegree_le {x : L} [FiniteDimensional K L] (hx : IsIntegral K x) :
- (minpoly K x).natDegree ≤ finrank K L :=
+theorem IntermediateField.minpoly.natDegree_le {x : L} [FiniteDimensional K L]
+ (hx : IsIntegral K x) : (minpoly K x).natDegree ≤ finrank K L :=
le_of_eq_of_le (IntermediateField.adjoin.finrank hx).symm K⟮⟯.toSubmodule.finrank_le
-#align minpoly.nat_degree_le minpoly.natDegree_le
+#align minpoly.nat_degree_le IntermediateField.minpoly.natDegree_le
-theorem minpoly.degree_le {x : L} [FiniteDimensional K L] (hx : IsIntegral K x) :
+theorem IntermediateField.minpoly.degree_le {x : L} [FiniteDimensional K L] (hx : IsIntegral K x) :
(minpoly K x).degree ≤ finrank K L :=
- degree_le_of_natDegree_le (minpoly.natDegree_le hx)
-#align minpoly.degree_le minpoly.degree_le
+ degree_le_of_natDegree_le (IntermediateField.minpoly.natDegree_le hx)
+#align minpoly.degree_le IntermediateField.minpoly.degree_le
end PowerBasis
@@ -931,15 +959,19 @@ section Induction
variable {F : Type _} [Field F] {E : Type _} [Field E] [Algebra F E]
+#print IntermediateField.FG /-
/-- An intermediate field `S` is finitely generated if there exists `t : finset E` such that
`intermediate_field.adjoin F t = S`. -/
-def Fg (S : IntermediateField F E) : Prop :=
+def FG (S : IntermediateField F E) : Prop :=
∃ t : Finset E, adjoin F ↑t = S
-#align intermediate_field.fg IntermediateField.Fg
+#align intermediate_field.fg IntermediateField.FG
+-/
+#print IntermediateField.fg_adjoin_finset /-
theorem fg_adjoin_finset (t : Finset E) : (adjoin F (↑t : Set E)).FG :=
⟨t, rfl⟩
#align intermediate_field.fg_adjoin_finset IntermediateField.fg_adjoin_finset
+-/
theorem fg_def {S : IntermediateField F E} : S.FG ↔ ∃ t : Set E, Set.Finite t ∧ adjoin F t = S :=
Iff.symm Set.exists_finite_iff_finset
@@ -993,10 +1025,12 @@ section AlgHomMkAdjoinSplits
variable (F E K : Type _) [Field F] [Field E] [Field K] [Algebra F E] [Algebra F K] {S : Set E}
+#print IntermediateField.Lifts /-
/-- Lifts `L → K` of `F → K` -/
def Lifts :=
Σ L : IntermediateField F E, L →ₐ[F] K
#align intermediate_field.lifts IntermediateField.Lifts
+-/
variable {F E K}
@@ -1115,6 +1149,7 @@ theorem Lifts.exists_upper_bound (c : Set (Lifts F E K)) (hc : IsChain (· ≤
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
+#print IntermediateField.Lifts.liftOfSplits /-
/-- Extend a lift `x : lifts F E K` to an element `s : E` whose conjugates are all in `K` -/
noncomputable def Lifts.liftOfSplits (x : Lifts F E K) {s : E} (h1 : IsIntegral F s)
(h2 : (minpoly F s).Splits (algebraMap F K)) : Lifts F E K :=
@@ -1133,6 +1168,7 @@ noncomputable def Lifts.liftOfSplits (x : Lifts F E K) {s : E} (h1 : IsIntegral
simp_rw [mem_roots (map_ne_zero (minpoly.ne_zero h3)), is_root, ← eval₂_eq_eval_map]
exact map_root_of_splits x.2.toRingHom key (ne_of_gt (minpoly.degree_pos h3))⟩⟩⟩
#align intermediate_field.lifts.lift_of_splits IntermediateField.Lifts.liftOfSplits
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
bump to nightly-2023-06-06-03
mathlib commit https://github.com/leanprover-community/mathlib/commit/a3209ddf94136d36e5e5c624b10b2a347cc9d090
Diff
@@ -4,13 +4,13 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
! This file was ported from Lean 3 source module field_theory.adjoin
-! leanprover-community/mathlib commit d4437c68c8d350fc9d4e95e1e174409db35e30d7
+! leanprover-community/mathlib commit df76f43357840485b9d04ed5dee5ab115d420e87
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
import Mathbin.FieldTheory.IntermediateField
import Mathbin.FieldTheory.Separable
-import Mathbin.FieldTheory.SplittingField
+import Mathbin.FieldTheory.SplittingField.IsSplittingField
import Mathbin.RingTheory.TensorProduct
/-!
bump to nightly-2023-06-04-18
mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297
Diff
@@ -454,7 +454,7 @@ instance (priority := 900) insertNonempty {α : Type _} (s : Set α) : Insert s
#align intermediate_field.insert_nonempty IntermediateField.insertNonempty
-- mathport name: «expr ⟮ ,⟯»
-notation3:max K"⟮"(l", "* => foldr (h t => Insert.insert t h) ∅)"⟯" => adjoin K l
+notation3:max K"⟮"(l ", "* => foldr (h t => Insert.insert t h) ∅)"⟯" => adjoin K l
section AdjoinSimple
bump to nightly-2023-06-01-16
mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb
Diff
@@ -176,7 +176,7 @@ theorem iInf_toSubfield {ι : Sort _} (S : ι → IntermediateField F E) :
def equivOfEq {S T : IntermediateField F E} (h : S = T) : S ≃ₐ[F] T := by
refine'
{ toFun := fun x => ⟨x, _⟩
- invFun := fun x => ⟨x, _⟩.. } <;>
+ invFun := fun x => ⟨x, _⟩ .. } <;>
tidy
#align intermediate_field.equiv_of_eq IntermediateField.equivOfEq
@@ -506,7 +506,7 @@ variable {F} {α}
theorem adjoin_algebraic_toSubalgebra {S : Set E} (hS : ∀ x ∈ S, IsAlgebraic F x) :
(IntermediateField.adjoin F S).toSubalgebra = Algebra.adjoin F S :=
by
- simp only [isAlgebraic_iff_isIntegral] at hS
+ simp only [isAlgebraic_iff_isIntegral] at hS
have : Algebra.IsIntegral F (Algebra.adjoin F S) := by
rwa [← le_integralClosure_iff_isIntegral, Algebra.adjoin_le_iff]
have := isField_of_isIntegral_of_isField' this (Field.toIsField F)
@@ -546,7 +546,7 @@ theorem isSplittingField_iSup {ι : Type _} {t : ι → IntermediateField F E} {
by
let K : IntermediateField F E := ⨆ i ∈ s, t i
have hK : ∀ i ∈ s, t i ≤ K := fun i hi => le_iSup_of_le i (le_iSup (fun _ => t i) hi)
- simp only [is_splitting_field_iff] at h⊢
+ simp only [is_splitting_field_iff] at h ⊢
refine'
⟨splits_prod (algebraMap F K) fun i hi =>
Polynomial.splits_comp_of_splits (algebraMap F (t i)) (inclusion (hK i hi)).toRingHom
@@ -570,7 +570,7 @@ theorem adjoin_simple_isCompactElement (x : E) : IsCompactElement F⟮⟯ :=
by
rw [is_compact_element_iff_le_of_directed_Sup_le]
rintro s ⟨F₀, hF₀⟩ hs hx
- simp only [adjoin_simple_le_iff] at hx⊢
+ simp only [adjoin_simple_le_iff] at hx ⊢
let F : IntermediateField F E :=
{ carrier := ⋃ E ∈ s, ↑E
add_mem' :=
@@ -629,13 +629,13 @@ theorem exists_finset_of_mem_iSup {ι : Type _} {f : ι → IntermediateField F
(hx : x ∈ ⨆ i, f i) : ∃ s : Finset ι, x ∈ ⨆ i ∈ s, f i :=
by
have := (adjoin_simple_is_compact_element x).exists_finset_of_le_iSup (IntermediateField F E) f
- simp only [adjoin_simple_le_iff] at this
+ simp only [adjoin_simple_le_iff] at this
exact this hx
#align intermediate_field.exists_finset_of_mem_supr IntermediateField.exists_finset_of_mem_iSup
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
theorem exists_finset_of_mem_supr' {ι : Type _} {f : ι → IntermediateField F E} {x : E}
- (hx : x ∈ ⨆ i, f i) : ∃ s : Finset (Σi, f i), x ∈ ⨆ i ∈ s, F⟮⟯ :=
+ (hx : x ∈ ⨆ i, f i) : ∃ s : Finset (Σ i, f i), x ∈ ⨆ i ∈ s, F⟮⟯ :=
exists_finset_of_mem_iSup
(SetLike.le_def.mp
(iSup_le fun i x h =>
@@ -645,7 +645,7 @@ theorem exists_finset_of_mem_supr' {ι : Type _} {f : ι → IntermediateField F
theorem exists_finset_of_mem_supr'' {ι : Type _} {f : ι → IntermediateField F E}
(h : ∀ i, Algebra.IsAlgebraic F (f i)) {x : E} (hx : x ∈ ⨆ i, f i) :
- ∃ s : Finset (Σi, f i), x ∈ ⨆ i ∈ s, adjoin F ((minpoly F (i.2 : _)).rootSet E) :=
+ ∃ s : Finset (Σ i, f i), x ∈ ⨆ i ∈ s, adjoin F ((minpoly F (i.2 : _)).rootSet E) :=
by
refine'
exists_finset_of_mem_supr
@@ -802,7 +802,7 @@ variable {K : Type _} [Field K] [Algebra F K]
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
theorem minpoly_gen {α : E} (h : IsIntegral F α) : minpoly F (AdjoinSimple.gen F α) = minpoly F α :=
by
- rw [← adjoin_simple.algebra_map_gen F α] at h
+ rw [← adjoin_simple.algebra_map_gen F α] at h
have inj := (algebraMap F⟮⟯ E).Injective
exact
minpoly.eq_of_algebraMap_eq inj ((isIntegral_algebraMap_iff inj).mp h)
@@ -995,7 +995,7 @@ variable (F E K : Type _) [Field F] [Field E] [Field K] [Algebra F E] [Algebra F
/-- Lifts `L → K` of `F → K` -/
def Lifts :=
- ΣL : IntermediateField F E, L →ₐ[F] K
+ Σ L : IntermediateField F E, L →ₐ[F] K
#align intermediate_field.lifts IntermediateField.Lifts
variable {F E K}
@@ -1166,14 +1166,14 @@ theorem algHom_mk_adjoin_splits
(fun s t => x.2.map_mul ⟨s, _⟩ ⟨t, _⟩) x.2.map_zero (fun s t => x.2.map_add ⟨s, _⟩ ⟨t, _⟩)
x.2.commutes⟩
rcases x.exists_lift_of_splits (hK s hs).1 (hK s hs).2 with ⟨y, h1, h2⟩
- rwa [hx y h1] at h2
+ rwa [hx y h1] at h2
#align intermediate_field.alg_hom_mk_adjoin_splits IntermediateField.algHom_mk_adjoin_splits
theorem algHom_mk_adjoin_splits' (hS : adjoin F S = ⊤)
(hK : ∀ x ∈ S, IsIntegral F (x : E) ∧ (minpoly F x).Splits (algebraMap F K)) :
Nonempty (E →ₐ[F] K) := by
cases' alg_hom_mk_adjoin_splits hK with ϕ
- rw [hS] at ϕ
+ rw [hS] at ϕ
exact ⟨ϕ.comp top_equiv.symm.to_alg_hom⟩
#align intermediate_field.alg_hom_mk_adjoin_splits' IntermediateField.algHom_mk_adjoin_splits'
@@ -1217,7 +1217,7 @@ instance finiteDimensional_sup [h1 : FiniteDimensional K E1] [h2 : FiniteDimensi
suffices g.range = (E1 ⊔ E2).toSubalgebra
by
have h : FiniteDimensional K g.range.to_submodule := g.to_linear_map.finite_dimensional_range
- rwa [this] at h
+ rwa [this] at h
rw [Algebra.TensorProduct.productMap_range, E1.range_val, E2.range_val, sup_to_subalgebra]
#align intermediate_field.finite_dimensional_sup IntermediateField.finiteDimensional_sup
@@ -1258,7 +1258,7 @@ theorem isAlgebraic_iSup {ι : Type _} {f : ι → IntermediateField K L}
rintro ⟨x, hx⟩
obtain ⟨s, hx⟩ := exists_finset_of_mem_supr' hx
rw [is_algebraic_iff, Subtype.coe_mk, ← Subtype.coe_mk x hx, ← is_algebraic_iff]
- haveI : ∀ i : Σi, f i, FiniteDimensional K K⟮⟯ := fun ⟨i, x⟩ =>
+ haveI : ∀ i : Σ i, f i, FiniteDimensional K K⟮⟯ := fun ⟨i, x⟩ =>
adjoin.finite_dimensional (is_integral_iff.1 (isAlgebraic_iff_isIntegral.1 (h i x)))
apply Algebra.isAlgebraic_of_finite
#align intermediate_field.is_algebraic_supr IntermediateField.isAlgebraic_iSup
bump to nightly-2023-05-28-18
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
Diff
@@ -34,7 +34,7 @@ For example, `algebra.adjoin K {x}` might not include `x⁻¹`.
open FiniteDimensional Polynomial
-open Classical Polynomial
+open scoped Classical Polynomial
namespace IntermediateField
@@ -537,7 +537,7 @@ theorem adjoin_rootSet_isSplittingField {p : F[X]} (hp : p.Splits (algebraMap F
isSplittingField_iff.mpr ⟨splits_of_splits hp fun x hx => subset_adjoin F (p.rootSet E) hx, rfl⟩
#align intermediate_field.adjoin_root_set_is_splitting_field IntermediateField.adjoin_rootSet_isSplittingField
-open BigOperators
+open scoped BigOperators
/-- A compositum of splitting fields is a splitting field -/
theorem isSplittingField_iSup {ι : Type _} {t : ι → IntermediateField F E} {p : ι → F[X]}
bump to nightly-2023-05-28-04
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
Diff
@@ -91,9 +91,7 @@ theorem mem_bot {x : E} : x ∈ (⊥ : IntermediateField F E) ↔ x ∈ Set.rang
#align intermediate_field.mem_bot IntermediateField.mem_bot
@[simp]
-theorem bot_toSubalgebra : (⊥ : IntermediateField F E).toSubalgebra = ⊥ :=
- by
- ext
+theorem bot_toSubalgebra : (⊥ : IntermediateField F E).toSubalgebra = ⊥ := by ext;
rw [mem_to_subalgebra, Algebra.mem_bot, mem_bot]
#align intermediate_field.bot_to_subalgebra IntermediateField.bot_toSubalgebra
@@ -189,10 +187,8 @@ theorem equivOfEq_symm {S T : IntermediateField F E} (h : S = T) :
#align intermediate_field.equiv_of_eq_symm IntermediateField.equivOfEq_symm
@[simp]
-theorem equivOfEq_rfl (S : IntermediateField F E) : equivOfEq (rfl : S = S) = AlgEquiv.refl :=
- by
- ext
- rfl
+theorem equivOfEq_rfl (S : IntermediateField F E) : equivOfEq (rfl : S = S) = AlgEquiv.refl := by
+ ext; rfl
#align intermediate_field.equiv_of_eq_rfl IntermediateField.equivOfEq_rfl
@[simp]
@@ -254,10 +250,7 @@ theorem topEquiv_symm_apply_coe (a : E) : ↑(topEquiv.symm a : (⊤ : Intermedi
@[simp]
theorem restrictScalars_bot_eq_self (K : IntermediateField F E) :
- (⊥ : IntermediateField K E).restrictScalars _ = K :=
- by
- ext
- rw [mem_restrict_scalars, mem_bot]
+ (⊥ : IntermediateField K E).restrictScalars _ = K := by ext; rw [mem_restrict_scalars, mem_bot];
exact set.ext_iff.mp Subtype.range_coe x
#align intermediate_field.restrict_scalars_bot_eq_self IntermediateField.restrictScalars_bot_eq_self
@@ -371,8 +364,7 @@ theorem adjoin_adjoin_left (T : Set E) :
rw [SetLike.ext'_iff]
change ↑(adjoin (adjoin F S) T) = _
apply Set.eq_of_subset_of_subset <;> rw [adjoin_subset_adjoin_iff] <;> constructor
- · rintro _ ⟨⟨x, hx⟩, rfl⟩
- exact adjoin.mono _ _ _ (Set.subset_union_left _ _) hx
+ · rintro _ ⟨⟨x, hx⟩, rfl⟩; exact adjoin.mono _ _ _ (Set.subset_union_left _ _) hx
· exact subset_adjoin_of_subset_right _ _ (Set.subset_union_right _ _)
· exact subset_adjoin_of_subset_left _ (adjoin.range_algebra_map_subset _ _)
· exact Set.union_subset (subset_adjoin_of_subset_left _ (subset_adjoin _ _)) (subset_adjoin _ _)
@@ -674,18 +666,14 @@ section AdjoinIntermediateFieldLattice
variable {F : Type _} [Field F] {E : Type _} [Field E] [Algebra F E] {α : E} {S : Set E}
@[simp]
-theorem adjoin_eq_bot_iff : adjoin F S = ⊥ ↔ S ⊆ (⊥ : IntermediateField F E) :=
- by
- rw [eq_bot_iff, adjoin_le_iff]
- rfl
+theorem adjoin_eq_bot_iff : adjoin F S = ⊥ ↔ S ⊆ (⊥ : IntermediateField F E) := by
+ rw [eq_bot_iff, adjoin_le_iff]; rfl
#align intermediate_field.adjoin_eq_bot_iff IntermediateField.adjoin_eq_bot_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
@[simp]
-theorem adjoin_simple_eq_bot_iff : F⟮⟯ = ⊥ ↔ α ∈ (⊥ : IntermediateField F E) :=
- by
- rw [adjoin_eq_bot_iff]
- exact Set.singleton_subset_iff
+theorem adjoin_simple_eq_bot_iff : F⟮⟯ = ⊥ ↔ α ∈ (⊥ : IntermediateField F E) := by
+ rw [adjoin_eq_bot_iff]; exact Set.singleton_subset_iff
#align intermediate_field.adjoin_simple_eq_bot_iff IntermediateField.adjoin_simple_eq_bot_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
@@ -744,9 +732,7 @@ theorem rank_adjoin_eq_one_iff : Module.rank F (adjoin F S) = 1 ↔ S ⊆ (⊥ :
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
theorem rank_adjoin_simple_eq_one_iff : Module.rank F F⟮⟯ = 1 ↔ α ∈ (⊥ : IntermediateField F E) :=
- by
- rw [rank_adjoin_eq_one_iff]
- exact Set.singleton_subset_iff
+ by rw [rank_adjoin_eq_one_iff]; exact Set.singleton_subset_iff
#align intermediate_field.rank_adjoin_simple_eq_one_iff IntermediateField.rank_adjoin_simple_eq_one_iff
theorem finrank_adjoin_eq_one_iff : finrank F (adjoin F S) = 1 ↔ S ⊆ (⊥ : IntermediateField F E) :=
@@ -754,10 +740,8 @@ theorem finrank_adjoin_eq_one_iff : finrank F (adjoin F S) = 1 ↔ S ⊆ (⊥ :
#align intermediate_field.finrank_adjoin_eq_one_iff IntermediateField.finrank_adjoin_eq_one_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
-theorem finrank_adjoin_simple_eq_one_iff : finrank F F⟮⟯ = 1 ↔ α ∈ (⊥ : IntermediateField F E) :=
- by
- rw [finrank_adjoin_eq_one_iff]
- exact Set.singleton_subset_iff
+theorem finrank_adjoin_simple_eq_one_iff : finrank F F⟮⟯ = 1 ↔ α ∈ (⊥ : IntermediateField F E) := by
+ rw [finrank_adjoin_eq_one_iff]; exact Set.singleton_subset_iff
#align intermediate_field.finrank_adjoin_simple_eq_one_iff IntermediateField.finrank_adjoin_simple_eq_one_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
@@ -854,13 +838,9 @@ noncomputable def adjoinRootEquivAdjoin (h : IsIntegral F α) : AdjoinRoot (minp
(Set.union_subset
(fun x hx =>
Exists.cases_on hx fun y hy =>
- ⟨y, by
- rw [RingHom.comp_apply, AdjoinRoot.lift_of]
- exact hy⟩)
+ ⟨y, by rw [RingHom.comp_apply, AdjoinRoot.lift_of]; exact hy⟩)
(set.singleton_subset_iff.mpr
- ⟨AdjoinRoot.root (minpoly F α),
- by
- rw [RingHom.comp_apply, AdjoinRoot.lift_root]
+ ⟨AdjoinRoot.root (minpoly F α), by rw [RingHom.comp_apply, AdjoinRoot.lift_root];
rfl⟩)))
#align intermediate_field.adjoin_root_equiv_adjoin IntermediateField.adjoinRootEquivAdjoin
@@ -1076,12 +1056,8 @@ def Lifts.upperBoundIntermediateField {c : Set (Lifts F E K)} (hc : IsChain (·
carrier s := ∃ x : Lifts F E K, x ∈ Insert.insert ⊥ c ∧ (s ∈ x.1 : Prop)
zero_mem' := ⟨⊥, Set.mem_insert ⊥ c, zero_mem ⊥⟩
one_mem' := ⟨⊥, Set.mem_insert ⊥ c, one_mem ⊥⟩
- neg_mem' := by
- rintro _ ⟨x, y, h⟩
- exact ⟨x, ⟨y, x.1.neg_mem h⟩⟩
- inv_mem' := by
- rintro _ ⟨x, y, h⟩
- exact ⟨x, ⟨y, x.1.inv_mem h⟩⟩
+ neg_mem' := by rintro _ ⟨x, y, h⟩; exact ⟨x, ⟨y, x.1.neg_mem h⟩⟩
+ inv_mem' := by rintro _ ⟨x, y, h⟩; exact ⟨x, ⟨y, x.1.inv_mem h⟩⟩
add_mem' := by
rintro _ _ ⟨x, hx, ha⟩ ⟨y, hy, hb⟩
obtain ⟨z, hz, hxz, hyz⟩ := lifts.exists_max_two hc hx hy
@@ -1145,10 +1121,7 @@ noncomputable def Lifts.liftOfSplits (x : Lifts F E K) {s : E} (h1 : IsIntegral
let h3 : IsIntegral x.1 s := isIntegral_of_isScalarTower h1
let key : (minpoly x.1 s).Splits x.2.toRingHom :=
splits_of_splits_of_dvd _ (map_ne_zero (minpoly.ne_zero h1))
- ((splits_map_iff _ _).mpr
- (by
- convert h2
- exact RingHom.ext fun y => x.2.commutes y))
+ ((splits_map_iff _ _).mpr (by convert h2; exact RingHom.ext fun y => x.2.commutes y))
(minpoly.dvd_map_of_isScalarTower _ _ _)
⟨x.1⟮⟯.restrictScalars F,
(@algHomEquivSigma F x.1 (x.1⟮⟯.restrictScalars F) K _ _ _ _ _ _ _
bump to nightly-2023-05-23-03
mathlib commit https://github.com/leanprover-community/mathlib/commit/75e7fca56381d056096ce5d05e938f63a6567828
Diff
@@ -517,7 +517,7 @@ theorem adjoin_algebraic_toSubalgebra {S : Set E} (hS : ∀ x ∈ S, IsAlgebraic
simp only [isAlgebraic_iff_isIntegral] at hS
have : Algebra.IsIntegral F (Algebra.adjoin F S) := by
rwa [← le_integralClosure_iff_isIntegral, Algebra.adjoin_le_iff]
- have := isField_of_isIntegral_of_is_field' this (Field.toIsField F)
+ have := isField_of_isIntegral_of_isField' this (Field.toIsField F)
rw [← ((Algebra.adjoin F S).toIntermediateField' this).eq_adjoin_of_eq_algebra_adjoin F S] <;> rfl
#align intermediate_field.adjoin_algebraic_to_subalgebra IntermediateField.adjoin_algebraic_toSubalgebra
@@ -1232,7 +1232,7 @@ theorem sup_toSubalgebra [h1 : FiniteDimensional K E1] [h2 : FiniteDimensional K
· obtain ⟨y, h⟩ := this.mul_inv_cancel hx'
exact (congr_arg (· ∈ S1 ⊔ S2) <| eq_inv_of_mul_eq_one_right <| subtype.ext_iff.mp h).mp y.2
exact
- isField_of_isIntegral_of_is_field'
+ isField_of_isIntegral_of_isField'
(is_integral_sup.mpr ⟨Algebra.isIntegral_of_finite K E1, Algebra.isIntegral_of_finite K E2⟩)
(Field.toIsField K)
#align intermediate_field.sup_to_subalgebra IntermediateField.sup_toSubalgebra
bump to nightly-2023-05-16-10
mathlib commit https://github.com/leanprover-community/mathlib/commit/0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8
Diff
@@ -957,26 +957,26 @@ def Fg (S : IntermediateField F E) : Prop :=
∃ t : Finset E, adjoin F ↑t = S
#align intermediate_field.fg IntermediateField.Fg
-theorem fg_adjoin_finset (t : Finset E) : (adjoin F (↑t : Set E)).Fg :=
+theorem fg_adjoin_finset (t : Finset E) : (adjoin F (↑t : Set E)).FG :=
⟨t, rfl⟩
#align intermediate_field.fg_adjoin_finset IntermediateField.fg_adjoin_finset
-theorem fg_def {S : IntermediateField F E} : S.Fg ↔ ∃ t : Set E, Set.Finite t ∧ adjoin F t = S :=
+theorem fg_def {S : IntermediateField F E} : S.FG ↔ ∃ t : Set E, Set.Finite t ∧ adjoin F t = S :=
Iff.symm Set.exists_finite_iff_finset
#align intermediate_field.fg_def IntermediateField.fg_def
-theorem fg_bot : (⊥ : IntermediateField F E).Fg :=
+theorem fg_bot : (⊥ : IntermediateField F E).FG :=
⟨∅, adjoin_empty F E⟩
#align intermediate_field.fg_bot IntermediateField.fg_bot
-theorem fg_of_fg_toSubalgebra (S : IntermediateField F E) (h : S.toSubalgebra.Fg) : S.Fg :=
+theorem fG_of_fG_toSubalgebra (S : IntermediateField F E) (h : S.toSubalgebra.FG) : S.FG :=
by
cases' h with t ht
exact ⟨t, (eq_adjoin_of_eq_algebra_adjoin _ _ _ ht.symm).symm⟩
-#align intermediate_field.fg_of_fg_to_subalgebra IntermediateField.fg_of_fg_toSubalgebra
+#align intermediate_field.fg_of_fg_to_subalgebra IntermediateField.fG_of_fG_toSubalgebra
-theorem fg_of_noetherian (S : IntermediateField F E) [IsNoetherian F E] : S.Fg :=
- S.fg_of_fg_toSubalgebra S.toSubalgebra.fg_of_noetherian
+theorem fg_of_noetherian (S : IntermediateField F E) [IsNoetherian F E] : S.FG :=
+ S.fG_of_fG_toSubalgebra S.toSubalgebra.fg_of_noetherian
#align intermediate_field.fg_of_noetherian IntermediateField.fg_of_noetherian
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
@@ -993,7 +993,7 @@ theorem induction_on_adjoin_finset (S : Finset E) (P : IntermediateField F E →
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
theorem induction_on_adjoin_fg (P : IntermediateField F E → Prop) (base : P ⊥)
(ih : ∀ (K : IntermediateField F E) (x : E), P K → P (K⟮⟯.restrictScalars F))
- (K : IntermediateField F E) (hK : K.Fg) : P K :=
+ (K : IntermediateField F E) (hK : K.FG) : P K :=
by
obtain ⟨S, rfl⟩ := hK
exact induction_on_adjoin_finset S P base fun K x _ hK => ih K x hK
bump to nightly-2023-05-14-08
mathlib commit https://github.com/leanprover-community/mathlib/commit/e3fb84046afd187b710170887195d50bada934ee
Diff
@@ -140,38 +140,38 @@ theorem inf_toSubfield (S T : IntermediateField F E) :
#align intermediate_field.inf_to_subfield IntermediateField.inf_toSubfield
@[simp, norm_cast]
-theorem coe_infₛ (S : Set (IntermediateField F E)) : (↑(infₛ S) : Set E) = infₛ (coe '' S) :=
+theorem coe_sInf (S : Set (IntermediateField F E)) : (↑(sInf S) : Set E) = sInf (coe '' S) :=
rfl
-#align intermediate_field.coe_Inf IntermediateField.coe_infₛ
+#align intermediate_field.coe_Inf IntermediateField.coe_sInf
@[simp]
-theorem infₛ_toSubalgebra (S : Set (IntermediateField F E)) :
- (infₛ S).toSubalgebra = infₛ (toSubalgebra '' S) :=
- SetLike.coe_injective <| by simp [Set.unionₛ_image]
-#align intermediate_field.Inf_to_subalgebra IntermediateField.infₛ_toSubalgebra
+theorem sInf_toSubalgebra (S : Set (IntermediateField F E)) :
+ (sInf S).toSubalgebra = sInf (toSubalgebra '' S) :=
+ SetLike.coe_injective <| by simp [Set.sUnion_image]
+#align intermediate_field.Inf_to_subalgebra IntermediateField.sInf_toSubalgebra
@[simp]
-theorem infₛ_toSubfield (S : Set (IntermediateField F E)) :
- (infₛ S).toSubfield = infₛ (toSubfield '' S) :=
- SetLike.coe_injective <| by simp [Set.unionₛ_image]
-#align intermediate_field.Inf_to_subfield IntermediateField.infₛ_toSubfield
+theorem sInf_toSubfield (S : Set (IntermediateField F E)) :
+ (sInf S).toSubfield = sInf (toSubfield '' S) :=
+ SetLike.coe_injective <| by simp [Set.sUnion_image]
+#align intermediate_field.Inf_to_subfield IntermediateField.sInf_toSubfield
@[simp, norm_cast]
-theorem coe_infᵢ {ι : Sort _} (S : ι → IntermediateField F E) : (↑(infᵢ S) : Set E) = ⋂ i, S i := by
- simp [infᵢ]
-#align intermediate_field.coe_infi IntermediateField.coe_infᵢ
+theorem coe_iInf {ι : Sort _} (S : ι → IntermediateField F E) : (↑(iInf S) : Set E) = ⋂ i, S i := by
+ simp [iInf]
+#align intermediate_field.coe_infi IntermediateField.coe_iInf
@[simp]
-theorem infᵢ_toSubalgebra {ι : Sort _} (S : ι → IntermediateField F E) :
- (infᵢ S).toSubalgebra = ⨅ i, (S i).toSubalgebra :=
- SetLike.coe_injective <| by simp [infᵢ]
-#align intermediate_field.infi_to_subalgebra IntermediateField.infᵢ_toSubalgebra
+theorem iInf_toSubalgebra {ι : Sort _} (S : ι → IntermediateField F E) :
+ (iInf S).toSubalgebra = ⨅ i, (S i).toSubalgebra :=
+ SetLike.coe_injective <| by simp [iInf]
+#align intermediate_field.infi_to_subalgebra IntermediateField.iInf_toSubalgebra
@[simp]
-theorem infᵢ_toSubfield {ι : Sort _} (S : ι → IntermediateField F E) :
- (infᵢ S).toSubfield = ⨅ i, (S i).toSubfield :=
- SetLike.coe_injective <| by simp [infᵢ]
-#align intermediate_field.infi_to_subfield IntermediateField.infᵢ_toSubfield
+theorem iInf_toSubfield {ι : Sort _} (S : ι → IntermediateField F E) :
+ (iInf S).toSubfield = ⨅ i, (S i).toSubfield :=
+ SetLike.coe_injective <| by simp [iInf]
+#align intermediate_field.infi_to_subfield IntermediateField.iInf_toSubfield
/-- Construct an algebra isomorphism from an equality of intermediate fields -/
@[simps apply]
@@ -548,21 +548,21 @@ theorem adjoin_rootSet_isSplittingField {p : F[X]} (hp : p.Splits (algebraMap F
open BigOperators
/-- A compositum of splitting fields is a splitting field -/
-theorem isSplittingField_supᵢ {ι : Type _} {t : ι → IntermediateField F E} {p : ι → F[X]}
+theorem isSplittingField_iSup {ι : Type _} {t : ι → IntermediateField F E} {p : ι → F[X]}
{s : Finset ι} (h0 : (∏ i in s, p i) ≠ 0) (h : ∀ i ∈ s, (p i).IsSplittingField F (t i)) :
(∏ i in s, p i).IsSplittingField F (⨆ i ∈ s, t i : IntermediateField F E) :=
by
let K : IntermediateField F E := ⨆ i ∈ s, t i
- have hK : ∀ i ∈ s, t i ≤ K := fun i hi => le_supᵢ_of_le i (le_supᵢ (fun _ => t i) hi)
+ have hK : ∀ i ∈ s, t i ≤ K := fun i hi => le_iSup_of_le i (le_iSup (fun _ => t i) hi)
simp only [is_splitting_field_iff] at h⊢
refine'
⟨splits_prod (algebraMap F K) fun i hi =>
Polynomial.splits_comp_of_splits (algebraMap F (t i)) (inclusion (hK i hi)).toRingHom
(h i hi).1,
_⟩
- simp only [root_set_prod p s h0, ← Set.supᵢ_eq_unionᵢ, (@gc F _ E _ _).l_supᵢ₂]
- exact supᵢ_congr fun i => supᵢ_congr fun hi => (h i hi).2
-#align intermediate_field.is_splitting_field_supr IntermediateField.isSplittingField_supᵢ
+ simp only [root_set_prod p s h0, ← Set.iSup_eq_iUnion, (@gc F _ E _ _).l_iSup₂]
+ exact iSup_congr fun i => iSup_congr fun hi => (h i hi).2
+#align intermediate_field.is_splitting_field_supr IntermediateField.isSplittingField_iSup
open Set CompleteLattice
@@ -599,7 +599,7 @@ theorem adjoin_simple_isCompactElement (x : E) : IsCompactElement F⟮⟯ :=
exact mem_Union_of_mem E (mem_Union_of_mem hE (E.inv_mem hx))
algebraMap_mem' := fun x =>
mem_Union_of_mem F₀ (mem_Union_of_mem hF₀ (F₀.algebra_map_mem x)) }
- have key : Sup s ≤ F := supₛ_le fun E hE => subset_Union_of_subset E (subset_Union _ hE)
+ have key : Sup s ≤ F := sSup_le fun E hE => subset_Union_of_subset E (subset_Union _ hE)
obtain ⟨-, ⟨E, rfl⟩, -, ⟨hE, rfl⟩, hx⟩ := key hx
exact ⟨E, hE, hx⟩
#align intermediate_field.adjoin_simple_is_compact_element IntermediateField.adjoin_simple_isCompactElement
@@ -612,9 +612,9 @@ theorem adjoin_finset_isCompactElement (S : Finset E) :
have key : adjoin F ↑S = ⨆ x ∈ S, F⟮⟯ :=
le_antisymm
(adjoin_le_iff.mpr fun x hx =>
- set_like.mem_coe.mpr (adjoin_simple_le_iff.mp (le_supᵢ_of_le x (le_supᵢ_of_le hx le_rfl))))
- (supᵢ_le fun x => supᵢ_le fun hx => adjoin_simple_le_iff.mpr (subset_adjoin F S hx))
- rw [key, ← Finset.sup_eq_supᵢ]
+ set_like.mem_coe.mpr (adjoin_simple_le_iff.mp (le_iSup_of_le x (le_iSup_of_le hx le_rfl))))
+ (iSup_le fun x => iSup_le fun hx => adjoin_simple_le_iff.mpr (subset_adjoin F S hx))
+ rw [key, ← Finset.sup_eq_iSup]
exact finset_sup_compact_of_compact S fun x hx => adjoin_simple_is_compact_element x
#align intermediate_field.adjoin_finset_is_compact_element IntermediateField.adjoin_finset_isCompactElement
@@ -629,25 +629,25 @@ instance : IsCompactlyGenerated (IntermediateField F E) :=
⟨fun s =>
⟨(fun x => F⟮⟯) '' s,
⟨by rintro t ⟨x, hx, rfl⟩ <;> exact adjoin_simple_is_compact_element x,
- supₛ_image.trans
- (le_antisymm (supᵢ_le fun i => supᵢ_le fun hi => adjoin_simple_le_iff.mpr hi) fun x hx =>
- adjoin_simple_le_iff.mp (le_supᵢ_of_le x (le_supᵢ_of_le hx le_rfl)))⟩⟩⟩
+ sSup_image.trans
+ (le_antisymm (iSup_le fun i => iSup_le fun hi => adjoin_simple_le_iff.mpr hi) fun x hx =>
+ adjoin_simple_le_iff.mp (le_iSup_of_le x (le_iSup_of_le hx le_rfl)))⟩⟩⟩
-theorem exists_finset_of_mem_supᵢ {ι : Type _} {f : ι → IntermediateField F E} {x : E}
+theorem exists_finset_of_mem_iSup {ι : Type _} {f : ι → IntermediateField F E} {x : E}
(hx : x ∈ ⨆ i, f i) : ∃ s : Finset ι, x ∈ ⨆ i ∈ s, f i :=
by
- have := (adjoin_simple_is_compact_element x).exists_finset_of_le_supᵢ (IntermediateField F E) f
+ have := (adjoin_simple_is_compact_element x).exists_finset_of_le_iSup (IntermediateField F E) f
simp only [adjoin_simple_le_iff] at this
exact this hx
-#align intermediate_field.exists_finset_of_mem_supr IntermediateField.exists_finset_of_mem_supᵢ
+#align intermediate_field.exists_finset_of_mem_supr IntermediateField.exists_finset_of_mem_iSup
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
theorem exists_finset_of_mem_supr' {ι : Type _} {f : ι → IntermediateField F E} {x : E}
(hx : x ∈ ⨆ i, f i) : ∃ s : Finset (Σi, f i), x ∈ ⨆ i ∈ s, F⟮⟯ :=
- exists_finset_of_mem_supᵢ
+ exists_finset_of_mem_iSup
(SetLike.le_def.mp
- (supᵢ_le fun i x h =>
- SetLike.le_def.mp (le_supᵢ_of_le ⟨i, x, h⟩ le_rfl) (mem_adjoin_simple_self F x))
+ (iSup_le fun i x h =>
+ SetLike.le_def.mp (le_iSup_of_le ⟨i, x, h⟩ le_rfl) (mem_adjoin_simple_self F x))
hx)
#align intermediate_field.exists_finset_of_mem_supr' IntermediateField.exists_finset_of_mem_supr'
@@ -658,8 +658,8 @@ theorem exists_finset_of_mem_supr'' {ι : Type _} {f : ι → IntermediateField
refine'
exists_finset_of_mem_supr
(set_like.le_def.mp
- (supᵢ_le fun i x hx =>
- set_like.le_def.mp (le_supᵢ_of_le ⟨i, x, hx⟩ le_rfl) (subset_adjoin F _ _))
+ (iSup_le fun i x hx =>
+ set_like.le_def.mp (le_iSup_of_le ⟨i, x, hx⟩ le_rfl) (subset_adjoin F _ _))
hx)
rw [IntermediateField.minpoly_eq, Subtype.coe_mk, mem_root_set_of_ne, minpoly.aeval]
exact minpoly.ne_zero (is_integral_iff.mp (is_algebraic_iff_is_integral.mp (h i ⟨x, hx⟩)))
@@ -1248,37 +1248,37 @@ instance finiteDimensional_sup [h1 : FiniteDimensional K E1] [h2 : FiniteDimensi
rw [Algebra.TensorProduct.productMap_range, E1.range_val, E2.range_val, sup_to_subalgebra]
#align intermediate_field.finite_dimensional_sup IntermediateField.finiteDimensional_sup
-instance finiteDimensional_supᵢ_of_finite {ι : Type _} {t : ι → IntermediateField K L}
+instance finiteDimensional_iSup_of_finite {ι : Type _} {t : ι → IntermediateField K L}
[h : Finite ι] [∀ i, FiniteDimensional K (t i)] :
FiniteDimensional K (⨆ i, t i : IntermediateField K L) :=
by
- rw [← supᵢ_univ]
+ rw [← iSup_univ]
let P : Set ι → Prop := fun s => FiniteDimensional K (⨆ i ∈ s, t i : IntermediateField K L)
change P Set.univ
apply Set.Finite.induction_on
· exact Set.finite_univ
all_goals dsimp only [P]
- · rw [supᵢ_emptyset]
+ · rw [iSup_emptyset]
exact (bot_equiv K L).symm.toLinearEquiv.FiniteDimensional
· intro _ s _ _ hs
- rw [supᵢ_insert]
+ rw [iSup_insert]
exact IntermediateField.finiteDimensional_sup _ _
-#align intermediate_field.finite_dimensional_supr_of_finite IntermediateField.finiteDimensional_supᵢ_of_finite
+#align intermediate_field.finite_dimensional_supr_of_finite IntermediateField.finiteDimensional_iSup_of_finite
-instance finiteDimensional_supᵢ_of_finset {ι : Type _} {f : ι → IntermediateField K L}
+instance finiteDimensional_iSup_of_finset {ι : Type _} {f : ι → IntermediateField K L}
{s : Finset ι} [h : ∀ i ∈ s, FiniteDimensional K (f i)] :
FiniteDimensional K (⨆ i ∈ s, f i : IntermediateField K L) :=
by
haveI : ∀ i : { i // i ∈ s }, FiniteDimensional K (f i) := fun i => h i i.2
have : (⨆ i ∈ s, f i) = ⨆ i : { i // i ∈ s }, f i :=
- le_antisymm (supᵢ_le fun i => supᵢ_le fun h => le_supᵢ (fun i : { i // i ∈ s } => f i) ⟨i, h⟩)
- (supᵢ_le fun i => le_supᵢ_of_le i (le_supᵢ_of_le i.2 le_rfl))
- exact this.symm ▸ IntermediateField.finiteDimensional_supᵢ_of_finite
-#align intermediate_field.finite_dimensional_supr_of_finset IntermediateField.finiteDimensional_supᵢ_of_finset
+ le_antisymm (iSup_le fun i => iSup_le fun h => le_iSup (fun i : { i // i ∈ s } => f i) ⟨i, h⟩)
+ (iSup_le fun i => le_iSup_of_le i (le_iSup_of_le i.2 le_rfl))
+ exact this.symm ▸ IntermediateField.finiteDimensional_iSup_of_finite
+#align intermediate_field.finite_dimensional_supr_of_finset IntermediateField.finiteDimensional_iSup_of_finset
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
/-- A compositum of algebraic extensions is algebraic -/
-theorem isAlgebraic_supᵢ {ι : Type _} {f : ι → IntermediateField K L}
+theorem isAlgebraic_iSup {ι : Type _} {f : ι → IntermediateField K L}
(h : ∀ i, Algebra.IsAlgebraic K (f i)) :
Algebra.IsAlgebraic K (⨆ i, f i : IntermediateField K L) :=
by
@@ -1288,7 +1288,7 @@ theorem isAlgebraic_supᵢ {ι : Type _} {f : ι → IntermediateField K L}
haveI : ∀ i : Σi, f i, FiniteDimensional K K⟮⟯ := fun ⟨i, x⟩ =>
adjoin.finite_dimensional (is_integral_iff.1 (isAlgebraic_iff_isIntegral.1 (h i x)))
apply Algebra.isAlgebraic_of_finite
-#align intermediate_field.is_algebraic_supr IntermediateField.isAlgebraic_supᵢ
+#align intermediate_field.is_algebraic_supr IntermediateField.isAlgebraic_iSup
end Supremum
bump to nightly-2023-05-09-02
mathlib commit https://github.com/leanprover-community/mathlib/commit/d4437c68c8d350fc9d4e95e1e174409db35e30d7
Diff
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
! This file was ported from Lean 3 source module field_theory.adjoin
-! leanprover-community/mathlib commit 039a089d2a4b93c761b234f3e5f5aeb752bac60f
+! leanprover-community/mathlib commit d4437c68c8d350fc9d4e95e1e174409db35e30d7
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
@@ -277,6 +277,17 @@ theorem AlgHom.map_fieldRange {K L : Type _} [Field K] [Field L] [Algebra F K] [
SetLike.ext' (Set.range_comp g f).symm
#align alg_hom.map_field_range AlgHom.map_fieldRange
+theorem AlgHom.fieldRange_eq_top {K : Type _} [Field K] [Algebra F K] {f : E →ₐ[F] K} :
+ f.fieldRange = ⊤ ↔ Function.Surjective f :=
+ SetLike.ext'_iff.trans Set.range_iff_surjective
+#align alg_hom.field_range_eq_top AlgHom.fieldRange_eq_top
+
+@[simp]
+theorem AlgEquiv.fieldRange_eq_top {K : Type _} [Field K] [Algebra F K] (f : E ≃ₐ[F] K) :
+ (f : E →ₐ[F] K).fieldRange = ⊤ :=
+ AlgHom.fieldRange_eq_top.mpr f.Surjective
+#align alg_equiv.field_range_eq_top AlgEquiv.fieldRange_eq_top
+
end Lattice
section AdjoinDef
bump to nightly-2023-04-06-02
mathlib commit https://github.com/leanprover-community/mathlib/commit/06a655b5fcfbda03502f9158bbf6c0f1400886f9
Diff
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
! This file was ported from Lean 3 source module field_theory.adjoin
-! leanprover-community/mathlib commit 3d32bf9cef95940e3fe1ca0dd2412e0f21579f46
+! leanprover-community/mathlib commit 039a089d2a4b93c761b234f3e5f5aeb752bac60f
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
@@ -23,7 +23,7 @@ For example, `algebra.adjoin K {x}` might not include `x⁻¹`.
## Main results
- `adjoin_adjoin_left`: adjoining S and then T is the same as adjoining `S ∪ T`.
-- `bot_eq_top_of_dim_adjoin_eq_one`: if `F⟮x⟯` has dimension `1` over `F` for every `x`
+- `bot_eq_top_of_rank_adjoin_eq_one`: if `F⟮x⟯` has dimension `1` over `F` for every `x`
in `E` then `F = E`
## Notation
@@ -701,16 +701,17 @@ theorem adjoin_nat (n : ℕ) : F⟮⟯ = ⊥ :=
adjoin_simple_eq_bot_iff.mpr (coe_nat_mem ⊥ n)
#align intermediate_field.adjoin_nat IntermediateField.adjoin_nat
-section AdjoinDim
+section AdjoinRank
open FiniteDimensional Module
variable {K L : IntermediateField F E}
@[simp]
-theorem dim_eq_one_iff : Module.rank F K = 1 ↔ K = ⊥ := by
- rw [← to_subalgebra_eq_iff, ← dim_eq_dim_subalgebra, Subalgebra.dim_eq_one_iff, bot_to_subalgebra]
-#align intermediate_field.dim_eq_one_iff IntermediateField.dim_eq_one_iff
+theorem rank_eq_one_iff : Module.rank F K = 1 ↔ K = ⊥ := by
+ rw [← to_subalgebra_eq_iff, ← rank_eq_rank_subalgebra, Subalgebra.rank_eq_one_iff,
+ bot_to_subalgebra]
+#align intermediate_field.rank_eq_one_iff IntermediateField.rank_eq_one_iff
@[simp]
theorem finrank_eq_one_iff : finrank F K = 1 ↔ K = ⊥ := by
@@ -719,23 +720,23 @@ theorem finrank_eq_one_iff : finrank F K = 1 ↔ K = ⊥ := by
#align intermediate_field.finrank_eq_one_iff IntermediateField.finrank_eq_one_iff
@[simp]
-theorem dim_bot : Module.rank F (⊥ : IntermediateField F E) = 1 := by rw [dim_eq_one_iff]
-#align intermediate_field.dim_bot IntermediateField.dim_bot
+theorem rank_bot : Module.rank F (⊥ : IntermediateField F E) = 1 := by rw [rank_eq_one_iff]
+#align intermediate_field.rank_bot IntermediateField.rank_bot
@[simp]
theorem finrank_bot : finrank F (⊥ : IntermediateField F E) = 1 := by rw [finrank_eq_one_iff]
#align intermediate_field.finrank_bot IntermediateField.finrank_bot
-theorem dim_adjoin_eq_one_iff : Module.rank F (adjoin F S) = 1 ↔ S ⊆ (⊥ : IntermediateField F E) :=
- Iff.trans dim_eq_one_iff adjoin_eq_bot_iff
-#align intermediate_field.dim_adjoin_eq_one_iff IntermediateField.dim_adjoin_eq_one_iff
+theorem rank_adjoin_eq_one_iff : Module.rank F (adjoin F S) = 1 ↔ S ⊆ (⊥ : IntermediateField F E) :=
+ Iff.trans rank_eq_one_iff adjoin_eq_bot_iff
+#align intermediate_field.rank_adjoin_eq_one_iff IntermediateField.rank_adjoin_eq_one_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
-theorem dim_adjoin_simple_eq_one_iff : Module.rank F F⟮⟯ = 1 ↔ α ∈ (⊥ : IntermediateField F E) :=
+theorem rank_adjoin_simple_eq_one_iff : Module.rank F F⟮⟯ = 1 ↔ α ∈ (⊥ : IntermediateField F E) :=
by
- rw [dim_adjoin_eq_one_iff]
+ rw [rank_adjoin_eq_one_iff]
exact Set.singleton_subset_iff
-#align intermediate_field.dim_adjoin_simple_eq_one_iff IntermediateField.dim_adjoin_simple_eq_one_iff
+#align intermediate_field.rank_adjoin_simple_eq_one_iff IntermediateField.rank_adjoin_simple_eq_one_iff
theorem finrank_adjoin_eq_one_iff : finrank F (adjoin F S) = 1 ↔ S ⊆ (⊥ : IntermediateField F E) :=
Iff.trans finrank_eq_one_iff adjoin_eq_bot_iff
@@ -750,12 +751,12 @@ theorem finrank_adjoin_simple_eq_one_iff : finrank F F⟮⟯ = 1 ↔ α ∈ (⊥
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
/-- If `F⟮x⟯` has dimension `1` over `F` for every `x ∈ E` then `F = E`. -/
-theorem bot_eq_top_of_dim_adjoin_eq_one (h : ∀ x : E, Module.rank F F⟮⟯ = 1) :
+theorem bot_eq_top_of_rank_adjoin_eq_one (h : ∀ x : E, Module.rank F F⟮⟯ = 1) :
(⊥ : IntermediateField F E) = ⊤ := by
ext
rw [iff_true_right IntermediateField.mem_top]
- exact dim_adjoin_simple_eq_one_iff.mp (h x)
-#align intermediate_field.bot_eq_top_of_dim_adjoin_eq_one IntermediateField.bot_eq_top_of_dim_adjoin_eq_one
+ exact rank_adjoin_simple_eq_one_iff.mp (h x)
+#align intermediate_field.bot_eq_top_of_rank_adjoin_eq_one IntermediateField.bot_eq_top_of_rank_adjoin_eq_one
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
theorem bot_eq_top_of_finrank_adjoin_eq_one (h : ∀ x : E, finrank F F⟮⟯ = 1) :
@@ -766,10 +767,10 @@ theorem bot_eq_top_of_finrank_adjoin_eq_one (h : ∀ x : E, finrank F F⟮⟯ =
#align intermediate_field.bot_eq_top_of_finrank_adjoin_eq_one IntermediateField.bot_eq_top_of_finrank_adjoin_eq_one
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
-theorem subsingleton_of_dim_adjoin_eq_one (h : ∀ x : E, Module.rank F F⟮⟯ = 1) :
+theorem subsingleton_of_rank_adjoin_eq_one (h : ∀ x : E, Module.rank F F⟮⟯ = 1) :
Subsingleton (IntermediateField F E) :=
- subsingleton_of_bot_eq_top (bot_eq_top_of_dim_adjoin_eq_one h)
-#align intermediate_field.subsingleton_of_dim_adjoin_eq_one IntermediateField.subsingleton_of_dim_adjoin_eq_one
+ subsingleton_of_bot_eq_top (bot_eq_top_of_rank_adjoin_eq_one h)
+#align intermediate_field.subsingleton_of_rank_adjoin_eq_one IntermediateField.subsingleton_of_rank_adjoin_eq_one
/- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
theorem subsingleton_of_finrank_adjoin_eq_one (h : ∀ x : E, finrank F F⟮⟯ = 1) :
@@ -793,7 +794,7 @@ theorem subsingleton_of_finrank_adjoin_le_one [FiniteDimensional F E]
subsingleton_of_bot_eq_top (bot_eq_top_of_finrank_adjoin_le_one h)
#align intermediate_field.subsingleton_of_finrank_adjoin_le_one IntermediateField.subsingleton_of_finrank_adjoin_le_one
-end AdjoinDim
+end AdjoinRank
end AdjoinIntermediateFieldLattice
bump to nightly-2023-02-21-16
mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc
Changes in mathlib4
mathlib3
mathlib4
fix: generalize index types of iSup to Sort (#12114)
This breaks a few simp proofs which were expecting these lemmas to apply to the data binders but not the prop binders.
Diff
@@ -1219,6 +1219,7 @@ theorem _root_.minpoly.degree_le (x : L) [FiniteDimensional K L] :
degree_le_of_natDegree_le (minpoly.natDegree_le x)
#align minpoly.degree_le minpoly.degree_le
+-- TODO: generalize to `Sort`
/-- A compositum of algebraic extensions is algebraic -/
theorem isAlgebraic_iSup {ι : Type*} {t : ι → IntermediateField K L}
(h : ∀ i, Algebra.IsAlgebraic K (t i)) :
chore(Algebra/Algebra): split Subalgebra.Basic (#12267)
This PR was supposed to be simultaneous with #12090 but I got ill last week.
This is based on seeing the import Algebra.Algebra.Subalgebra.Basic → RingTheory.Ideal.Operations on the longest pole. It feels like Ideal.Operations should not be needed to define the notion of subalgebra, only to construct some interesting examples. So I removed the import and split off anything that wouldn't fit.
The following results and their corollaries were split off:
Subalgebra.prodSubalgebra.iSupLiftAlgHom.ker_rangeRestrictSubalgebra.mem_of_finset_sum_eq_one_of_pow_smul_mem
Co-authored-by: Anne Baanen <Vierkantor@users.noreply.github.com>
Diff
@@ -3,6 +3,7 @@ Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
+import Mathlib.Algebra.Algebra.Subalgebra.Directed
import Mathlib.FieldTheory.IntermediateField
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.SplittingField.IsSplittingField
Diff
@@ -1092,7 +1092,7 @@ noncomputable def adjoinRootEquivAdjoin (h : IsIntegral F α) :
· exact RingHom.injective f
· suffices F⟮α⟯.toSubfield ≤ RingHom.fieldRange (F⟮α⟯.toSubfield.subtype.comp f) by
intro x
- obtain ⟨y, hy⟩ := (this (Subtype.mem x))
+ obtain ⟨y, hy⟩ := this (Subtype.mem x)
exact ⟨y, Subtype.ext hy⟩
refine' Subfield.closure_le.mpr (Set.union_subset (fun x hx => _) _)
· obtain ⟨y, hy⟩ := hx
feat: add Subalgebra.finite_(bot|sup) (#12025)
... and deprecated Subalgebra.finiteDimensional_(bot|sup)
Diff
@@ -1194,7 +1194,7 @@ theorem exists_lt_finrank_of_infinite_dimensional
(halg : Algebra.IsAlgebraic F E) (hnfd : ¬ FiniteDimensional F E) (n : ℕ) :
∃ L : IntermediateField F E, FiniteDimensional F L ∧ n < finrank F L := by
induction' n with n ih
- · exact ⟨⊥, Subalgebra.finiteDimensional_bot, finrank_pos⟩
+ · exact ⟨⊥, Subalgebra.finite_bot, finrank_pos⟩
obtain ⟨L, fin, hn⟩ := ih
obtain ⟨x, hx⟩ : ∃ x : E, x ∉ L := by
contrapose! hnfd
Diff
@@ -945,14 +945,18 @@ theorem adjoin_one : F⟮(1 : E)⟯ = ⊥ :=
#align intermediate_field.adjoin_one IntermediateField.adjoin_one
@[simp]
-theorem adjoin_int (n : ℤ) : F⟮(n : E)⟯ = ⊥ := by
- exact adjoin_simple_eq_bot_iff.mpr (coe_int_mem ⊥ n)
-#align intermediate_field.adjoin_int IntermediateField.adjoin_int
+theorem adjoin_intCast (n : ℤ) : F⟮(n : E)⟯ = ⊥ := by
+ exact adjoin_simple_eq_bot_iff.mpr (intCast_mem ⊥ n)
+#align intermediate_field.adjoin_int IntermediateField.adjoin_intCast
@[simp]
-theorem adjoin_nat (n : ℕ) : F⟮(n : E)⟯ = ⊥ :=
- adjoin_simple_eq_bot_iff.mpr (coe_nat_mem ⊥ n)
-#align intermediate_field.adjoin_nat IntermediateField.adjoin_nat
+theorem adjoin_natCast (n : ℕ) : F⟮(n : E)⟯ = ⊥ :=
+ adjoin_simple_eq_bot_iff.mpr (natCast_mem ⊥ n)
+#align intermediate_field.adjoin_nat IntermediateField.adjoin_natCast
+
+-- 2024-04-05
+@[deprecated] alias adjoin_int := adjoin_intCast
+@[deprecated] alias adjoin_nat := adjoin_natCast
section AdjoinRank
style: replace '.-/' by '. -/' (#11938)
Purely automatic replacement. If this is in any way controversial; I'm happy to just close this PR.
Diff
@@ -753,7 +753,7 @@ instance finiteDimensional_iSup_of_finite [h : Finite ι] [∀ i, FiniteDimensio
instance finiteDimensional_iSup_of_finset
/- Porting note: changed `h` from `∀ i ∈ s, FiniteDimensional K (t i)` because this caused an
error. See `finiteDimensional_iSup_of_finset'` for a stronger version, that was the one
- used in mathlib3.-/
+ used in mathlib3. -/
{s : Finset ι} [∀ i, FiniteDimensional K (t i)] :
FiniteDimensional K (⨆ i ∈ s, t i : IntermediateField K L) :=
iSup_subtype'' s t ▸ IntermediateField.finiteDimensional_iSup_of_finite
@@ -761,7 +761,7 @@ instance finiteDimensional_iSup_of_finset
theorem finiteDimensional_iSup_of_finset'
/- Porting note: this was the mathlib3 version. Using `[h : ...]`, as in mathlib3, causes the
- error "invalid parametric local instance".-/
+ error "invalid parametric local instance". -/
{s : Finset ι} (h : ∀ i ∈ s, FiniteDimensional K (t i)) :
FiniteDimensional K (⨆ i ∈ s, t i : IntermediateField K L) :=
have := Subtype.forall'.mp h
style: replace '/--A' by '/-- A' for each letter A. (#11939)
Also do the same for "/-A". This is a purely aesthetic change (and exhaustive).
Diff
@@ -751,7 +751,7 @@ instance finiteDimensional_iSup_of_finite [h : Finite ι] [∀ i, FiniteDimensio
#align intermediate_field.finite_dimensional_supr_of_finite IntermediateField.finiteDimensional_iSup_of_finite
instance finiteDimensional_iSup_of_finset
- /-Porting note: changed `h` from `∀ i ∈ s, FiniteDimensional K (t i)` because this caused an
+ /- Porting note: changed `h` from `∀ i ∈ s, FiniteDimensional K (t i)` because this caused an
error. See `finiteDimensional_iSup_of_finset'` for a stronger version, that was the one
used in mathlib3.-/
{s : Finset ι} [∀ i, FiniteDimensional K (t i)] :
@@ -760,7 +760,7 @@ instance finiteDimensional_iSup_of_finset
#align intermediate_field.finite_dimensional_supr_of_finset IntermediateField.finiteDimensional_iSup_of_finset
theorem finiteDimensional_iSup_of_finset'
- /-Porting note: this was the mathlib3 version. Using `[h : ...]`, as in mathlib3, causes the
+ /- Porting note: this was the mathlib3 version. Using `[h : ...]`, as in mathlib3, causes the
error "invalid parametric local instance".-/
{s : Finset ι} (h : ∀ i ∈ s, FiniteDimensional K (t i)) :
FiniteDimensional K (⨆ i ∈ s, t i : IntermediateField K L) :=
chore(*): remove empty lines between variable statements (#11418)
Empty lines were removed by executing the following Python script twice
import os
import re
# Loop through each file in the repository
for dir_path, dirs, files in os.walk('.'):
for filename in files:
if filename.endswith('.lean'):
file_path = os.path.join(dir_path, filename)
# Open the file and read its contents
with open(file_path, 'r') as file:
content = file.read()
# Use a regular expression to replace sequences of "variable" lines separated by empty lines
# with sequences without empty lines
modified_content = re.sub(r'(variable.*\n)\n(variable(?! .* in))', r'\1\2', content)
# Write the modified content back to the file
with open(file_path, 'w') as file:
file.write(modified_content)
Diff
@@ -788,7 +788,6 @@ end Supremum
section Tower
variable (E)
-
variable {K : Type*} [Field K] [Algebra F K] [Algebra E K] [IsScalarTower F E K]
/-- If `K / E / F` is a field extension tower, `L` is an intermediate field of `K / F`, such that
@@ -1062,7 +1061,6 @@ section AdjoinIntegralElement
universe u
variable (F : Type*) [Field F] {E : Type*} [Field E] [Algebra F E] {α : E}
-
variable {K : Type u} [Field K] [Algebra F K]
theorem minpoly_gen (α : E) :
chore(Algebra): improve argument names to induction principles (#11439)
These are the case names used by the induction tactic after the with.
This replaces H0, H1, Hmul etc with zero, one, mul.
This PR does not touch Submonoid or Subgroup, as to_additive does not know how to rename the argument names.
There are ways to work around this, but I'd prefer to leave them to a later PR.
This also leaves the closure_induction₂ variants alone, as renaming the arguments is more work for less gain.
Diff
@@ -536,12 +536,13 @@ theorem adjoin_eq_top_of_algebra (hS : Algebra.adjoin F S = ⊤) : adjoin F S =
top_le_iff.mp (hS.symm.trans_le <| algebra_adjoin_le_adjoin F S)
@[elab_as_elim]
-theorem adjoin_induction {s : Set E} {p : E → Prop} {x} (h : x ∈ adjoin F s) (Hs : ∀ x ∈ s, p x)
- (Hmap : ∀ x, p (algebraMap F E x)) (Hadd : ∀ x y, p x → p y → p (x + y))
- (Hneg : ∀ x, p x → p (-x)) (Hinv : ∀ x, p x → p x⁻¹) (Hmul : ∀ x y, p x → p y → p (x * y)) :
+theorem adjoin_induction {s : Set E} {p : E → Prop} {x} (h : x ∈ adjoin F s) (mem : ∀ x ∈ s, p x)
+ (algebraMap : ∀ x, p (algebraMap F E x)) (add : ∀ x y, p x → p y → p (x + y))
+ (neg : ∀ x, p x → p (-x)) (inv : ∀ x, p x → p x⁻¹) (mul : ∀ x y, p x → p y → p (x * y)) :
p x :=
- Subfield.closure_induction h (fun x hx => Or.casesOn hx (fun ⟨x, hx⟩ => hx ▸ Hmap x) (Hs x))
- ((algebraMap F E).map_one ▸ Hmap 1) Hadd Hneg Hinv Hmul
+ Subfield.closure_induction h
+ (fun x hx => Or.casesOn hx (fun ⟨x, hx⟩ => hx ▸ algebraMap x) (mem x))
+ ((_root_.algebraMap F E).map_one ▸ algebraMap 1) add neg inv mul
#align intermediate_field.adjoin_induction IntermediateField.adjoin_induction
/- Porting note (kmill): this notation is replacing the typeclass-based one I had previously
Acl/reorg tensor product (#11282)
Move:
Mathlib/Algebra/Module/DirectLimitAndTensorProduct.leantoLinearAlgebra/TensorProduct/DirectLimit.leanMathlib/LinearAlgebra/TensorProducttoMathlib/LinearAlgebra.TensorProduct.Basic.leanMathlib/RingTheory/TensorProducttoMathlib/RingTheory/TensorProduct/Basic.lean.
This follows suggestions 1, 2, 3 of
Co-authored-by: Antoine Chambert-Loir <antoine.chambert-loir@math.univ-paris-diderot.fr>
Diff
@@ -6,7 +6,7 @@ Authors: Thomas Browning, Patrick Lutz
import Mathlib.FieldTheory.IntermediateField
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.SplittingField.IsSplittingField
-import Mathlib.RingTheory.TensorProduct
+import Mathlib.RingTheory.TensorProduct.Basic
#align_import field_theory.adjoin from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
feat: sum and product of commuting semisimple endomorphisms (#10808)
-
Prove
isSemisimple_of_mem_adjoin: if two commuting endomorphisms of a finite-dimensional vector space over a perfect field are both semisimple, then every endomorphism in the algebra generated by them (in particular their product and sum) is semisimple. -
In the same file LinearAlgebra/Semisimple.lean,
eq_zero_of_isNilpotent_isSemisimpleandisSemisimple_of_squarefree_aeval_eq_zeroare golfed, andIsSemisimple.minpoly_squarefreeis proved
RingTheory/SimpleModule.lean:
-
Define
IsSemisimpleRing Rto mean that R is a semisimple R-module. add properties of simple modules and a characterization (they are exactly the quotients of the ring by maximal left ideals). -
The annihilator of a semisimple module is a radical ideal.
-
Any module over a semisimple ring is semisimple.
-
A finite product of semisimple rings is semisimple.
-
Any quotient of a semisimple ring is semisimple.
-
Add Artin--Wedderburn as a TODO (proof_wanted).
-
Order/Atoms.lean: add the instance from
IsSimpleOrdertoComplementedLattice, so thatIsSimpleModule → IsSemisimpleModuleis automatically inferred.
Prerequisites for showing a product of semisimple rings is semisimple:
-
Algebra/Module/Submodule/Map.lean: generalize
orderIsoMapComapso that it only requiresRingHomSurjectiverather thanRingHomInvPair -
Algebra/Ring/CompTypeclasses.lean, Mathlib/Algebra/Ring/Pi.lean, Algebra/Ring/Prod.lean: add RingHomSurjective instances
RingTheory/Artinian.lean:
-
quotNilradicalEquivPi: the quotient of a commutative Artinian ring R by its nilradical is isomorphic to the (finite) product of its quotients by maximal ideals (therefore a product of fields).equivPi: if the ring is moreover reduced, then the ring itself is a product of fields. Deduce that R is a semisimple ring and both R and R[X] are decomposition monoids. RequiresRingEquiv.quotientBotin RingTheory/Ideal/QuotientOperations.lean. -
Data/Polynomial/Eval.lean: the polynomial ring over a finite product of rings is isomorphic to the product of polynomial rings over individual rings. (Used to show R[X] is a decomposition monoid.)
Other necessary results:
-
FieldTheory/Minpoly/Field.lean: the minimal polynomial of an element in a reduced algebra over a field is radical.
-
RingTheory/PowerBasis.lean: generalize
PowerBasis.finiteDimensionaland rename it to.finite.
Annihilator stuff, some of which do not end up being used:
-
RingTheory/Ideal/Operations.lean: define
Module.annihilatorand redefineSubmodule.annihilatorin terms of it; add lemmas, including one that says an arbitrary intersection of radical ideals is radical. The new lemmaIdeal.isRadical_iff_pow_one_ltdepends onpow_imp_self_of_one_ltin Mathlib/Data/Nat/Interval.lean, which is also used to golf the proof ofisRadical_iff_pow_one_lt. -
Algebra/Module/Torsion.lean: add a lemma and an instance (unused)
-
Data/Polynomial/Module/Basic.lean: add a def (unused) and a lemma
-
LinearAlgebra/AnnihilatingPolynomial.lean: add lemma
span_minpoly_eq_annihilator
Some results about idempotent linear maps (projections) and idempotent elements, used to show that any (left) ideal in a semisimple ring is spanned by an idempotent element (unused):
-
LinearAlgebra/Projection.lean: add def
isIdempotentElemEquiv -
LinearAlgebra/Span.lean: add two lemmas
Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>
Diff
@@ -1136,7 +1136,7 @@ noncomputable def adjoin.powerBasis {x : L} (hx : IsIntegral K x) : PowerBasis K
#align intermediate_field.adjoin.power_basis IntermediateField.adjoin.powerBasis
theorem adjoin.finiteDimensional {x : L} (hx : IsIntegral K x) : FiniteDimensional K K⟮x⟯ :=
- PowerBasis.finiteDimensional (adjoin.powerBasis hx)
+ (adjoin.powerBasis hx).finite
#align intermediate_field.adjoin.finite_dimensional IntermediateField.adjoin.finiteDimensional
theorem isAlgebraic_adjoin_simple {x : L} (hx : IsIntegral K x) : Algebra.IsAlgebraic K K⟮x⟯ :=
@@ -1297,7 +1297,7 @@ theorem _root_.Polynomial.irreducible_comp {f g : K[X]} (hfm : f.Monic) (hgm : g
(this.trans natDegree_comp.symm).ge).irreducible hp₁
have := Fact.mk hp₁
let Kx := AdjoinRoot p
- letI := (AdjoinRoot.powerBasis hp₁.ne_zero).finiteDimensional
+ letI := (AdjoinRoot.powerBasis hp₁.ne_zero).finite
have key₁ : f = minpoly K (aeval (root p) g) := by
refine minpoly.eq_of_irreducible_of_monic hf ?_ hfm
rw [← aeval_comp]
style: homogenise porting notes (#11145)
Homogenises porting notes via capitalisation and addition of whitespace.
It makes the following changes:
- converts "--porting note" into "-- Porting note";
- converts "porting note" into "Porting note".
Diff
@@ -41,7 +41,7 @@ section AdjoinDef
variable (F : Type*) [Field F] {E : Type*} [Field E] [Algebra F E] (S : Set E)
---Porting note: not adding `neg_mem'` causes an error.
+-- Porting note: not adding `neg_mem'` causes an error.
/-- `adjoin F S` extends a field `F` by adjoining a set `S ⊆ E`. -/
def adjoin : IntermediateField F E :=
{ Subfield.closure (Set.range (algebraMap F E) ∪ S) with
@@ -1076,7 +1076,7 @@ theorem aeval_gen_minpoly (α : E) : aeval (AdjoinSimple.gen F α) (minpoly F α
exact (aeval_algebraMap_apply E (AdjoinSimple.gen F α) _).symm
#align intermediate_field.aeval_gen_minpoly IntermediateField.aeval_gen_minpoly
---Porting note: original proof used `Exists.cases_on`.
+-- Porting note: original proof used `Exists.cases_on`.
/-- algebra isomorphism between `AdjoinRoot` and `F⟮α⟯` -/
noncomputable def adjoinRootEquivAdjoin (h : IsIntegral F α) :
AdjoinRoot (minpoly F α) ≃ₐ[F] F⟮α⟯ :=
@@ -1343,7 +1343,7 @@ theorem fg_def {S : IntermediateField F E} : S.FG ↔ ∃ t : Set E, Set.Finite
#align intermediate_field.fg_def IntermediateField.fg_def
theorem fg_bot : (⊥ : IntermediateField F E).FG :=
- -- porting note: was `⟨∅, adjoin_empty F E⟩`
+ -- Porting note: was `⟨∅, adjoin_empty F E⟩`
⟨∅, by simp only [Finset.coe_empty, adjoin_empty]⟩
#align intermediate_field.fg_bot IntermediateField.fg_bot
Diff
@@ -740,7 +740,7 @@ instance finiteDimensional_iSup_of_finite [h : Finite ι] [∀ i, FiniteDimensio
let P : Set ι → Prop := fun s => FiniteDimensional K (⨆ i ∈ s, t i : IntermediateField K L)
change P Set.univ
apply Set.Finite.induction_on
- all_goals dsimp only
+ all_goals dsimp only [P]
· exact Set.finite_univ
· rw [iSup_emptyset]
exact (botEquiv K L).symm.toLinearEquiv.finiteDimensional
Diff
@@ -980,6 +980,12 @@ theorem rank_bot : Module.rank F (⊥ : IntermediateField F E) = 1 := by rw [ran
theorem finrank_bot : finrank F (⊥ : IntermediateField F E) = 1 := by rw [finrank_eq_one_iff]
#align intermediate_field.finrank_bot IntermediateField.finrank_bot
+@[simp] theorem rank_bot' : Module.rank (⊥ : IntermediateField F E) E = Module.rank F E := by
+ rw [← rank_mul_rank F (⊥ : IntermediateField F E) E, IntermediateField.rank_bot, one_mul]
+
+@[simp] theorem finrank_bot' : finrank (⊥ : IntermediateField F E) E = finrank F E :=
+ congr(Cardinal.toNat $(rank_bot'))
+
@[simp] protected theorem rank_top : Module.rank (⊤ : IntermediateField F E) E = 1 :=
Subalgebra.bot_eq_top_iff_rank_eq_one.mp <| top_le_iff.mp fun x _ ↦ ⟨⟨x, trivial⟩, rfl⟩
chore: remove terminal, terminal refines (#10762)
I replaced a few "terminal" refine/refine's with exact.
The strategy was very simple-minded: essentially any refine whose following line had smaller indentation got replaced by exact and then I cleaned up the mess.
This PR certainly leaves some further terminal refines, but maybe the current change is beneficial.
Diff
@@ -946,7 +946,7 @@ theorem adjoin_one : F⟮(1 : E)⟯ = ⊥ :=
@[simp]
theorem adjoin_int (n : ℤ) : F⟮(n : E)⟯ = ⊥ := by
- refine' adjoin_simple_eq_bot_iff.mpr (coe_int_mem ⊥ n)
+ exact adjoin_simple_eq_bot_iff.mpr (coe_int_mem ⊥ n)
#align intermediate_field.adjoin_int IntermediateField.adjoin_int
@[simp]
chore: remove stream-of-consciousness uses of have, replace and suffices (#10640)
No changes to tactic file, it's just boring fixes throughout the library.
This follows on from #6964.
Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>
Diff
@@ -1163,12 +1163,12 @@ theorem adjoin_minpoly_coeff_of_exists_primitive_element
rintro ⟨n, -, rfl⟩
rw [coeff_map]
apply Subtype.mem
- have dvd_g : minpoly K' α ∣ g.toSubring K'.toSubring (subset_adjoin F _)
- · apply minpoly.dvd
+ have dvd_g : minpoly K' α ∣ g.toSubring K'.toSubring (subset_adjoin F _) := by
+ apply minpoly.dvd
erw [aeval_def, eval₂_eq_eval_map, g.map_toSubring K'.toSubring, eval_map, ← aeval_def]
exact minpoly.aeval K α
- have finrank_eq : ∀ K : IntermediateField F E, finrank K E = natDegree (minpoly K α)
- · intro K
+ have finrank_eq : ∀ K : IntermediateField F E, finrank K E = natDegree (minpoly K α) := by
+ intro K
have := adjoin.finrank (.of_finite K α)
erw [adjoin_eq_top_of_adjoin_eq_top F hprim, finrank_top K E] at this
exact this
@@ -1275,8 +1275,8 @@ theorem _root_.Polynomial.irreducible_comp {f g : K[X]} (hfm : f.Monic) (hgm : g
Irreducible (f.comp g) := by
have hf' : natDegree f ≠ 0 :=
fun e ↦ not_irreducible_C (f.coeff 0) (eq_C_of_natDegree_eq_zero e ▸ hf)
- have hg' : natDegree g ≠ 0
- · have := Fact.mk hf
+ have hg' : natDegree g ≠ 0 := by
+ have := Fact.mk hf
intro e
apply not_irreducible_C ((g.map (algebraMap _ _)).coeff 0 - AdjoinSimple.gen K (root f))
-- Needed to specialize `map_sub` to avoid a timeout #8386
@@ -1292,21 +1292,21 @@ theorem _root_.Polynomial.irreducible_comp {f g : K[X]} (hfm : f.Monic) (hgm : g
have := Fact.mk hp₁
let Kx := AdjoinRoot p
letI := (AdjoinRoot.powerBasis hp₁.ne_zero).finiteDimensional
- have key₁ : f = minpoly K (aeval (root p) g)
- · refine minpoly.eq_of_irreducible_of_monic hf ?_ hfm
+ have key₁ : f = minpoly K (aeval (root p) g) := by
+ refine minpoly.eq_of_irreducible_of_monic hf ?_ hfm
rw [← aeval_comp]
exact aeval_eq_zero_of_dvd_aeval_eq_zero hp₂ (AdjoinRoot.eval₂_root p)
- have key₁' : finrank K K⟮aeval (root p) g⟯ = natDegree f
- · rw [adjoin.finrank, ← key₁]
+ have key₁' : finrank K K⟮aeval (root p) g⟯ = natDegree f := by
+ rw [adjoin.finrank, ← key₁]
exact IsIntegral.of_finite _ _
have key₂ : g.map (algebraMap _ _) - C (AdjoinSimple.gen K (aeval (root p) g)) =
- minpoly K⟮aeval (root p) g⟯ (root p)
- · exact minpoly.eq_of_irreducible_of_monic (hg _ _ key₁.symm) (by simp [AdjoinSimple.gen])
+ minpoly K⟮aeval (root p) g⟯ (root p) :=
+ minpoly.eq_of_irreducible_of_monic (hg _ _ key₁.symm) (by simp [AdjoinSimple.gen])
(Monic.sub_of_left (hgm.map _) (degree_lt_degree (by simpa [Nat.pos_iff_ne_zero] using hg')))
- have key₂' : finrank K⟮aeval (root p) g⟯ Kx = natDegree g
- · trans natDegree (minpoly K⟮aeval (root p) g⟯ (root p))
- · have : K⟮aeval (root p) g⟯⟮root p⟯ = ⊤
- · apply restrictScalars_injective K
+ have key₂' : finrank K⟮aeval (root p) g⟯ Kx = natDegree g := by
+ trans natDegree (minpoly K⟮aeval (root p) g⟯ (root p))
+ · have : K⟮aeval (root p) g⟯⟮root p⟯ = ⊤ := by
+ apply restrictScalars_injective K
rw [restrictScalars_top, adjoin_adjoin_left, Set.union_comm, ← adjoin_adjoin_left,
adjoin_root_eq_top p, restrictScalars_adjoin]
simp
Diff
@@ -761,7 +761,7 @@ instance finiteDimensional_iSup_of_finset
theorem finiteDimensional_iSup_of_finset'
/-Porting note: this was the mathlib3 version. Using `[h : ...]`, as in mathlib3, causes the
error "invalid parametric local instance".-/
- {s : Finset ι} (h : ∀ i, i ∈ s → FiniteDimensional K (t i)) :
+ {s : Finset ι} (h : ∀ i ∈ s, FiniteDimensional K (t i)) :
FiniteDimensional K (⨆ i ∈ s, t i : IntermediateField K L) :=
have := Subtype.forall'.mp h
iSup_subtype'' s t ▸ IntermediateField.finiteDimensional_iSup_of_finite
refactor(Data/FunLike): use unbundled inheritance from FunLike (#8386)
The FunLike hierarchy is very big and gets scanned through each time we need a coercion (via the CoeFun instance). It looks like unbundled inheritance suits Lean 4 better here. The only class that still extends FunLike is EquivLike, since that has a custom coe_injective' field that is easier to implement. All other classes should take FunLike or EquivLike as a parameter.
Important changes
Previously, morphism classes would be Type-valued and extend FunLike:
/-- `MyHomClass F A B` states that `F` is a type of `MyClass.op`-preserving morphisms.
You should extend this class when you extend `MyHom`. -/
class MyHomClass (F : Type*) (A B : outParam <| Type*) [MyClass A] [MyClass B]
extends FunLike F A B :=
(map_op : ∀ (f : F) (x y : A), f (MyClass.op x y) = MyClass.op (f x) (f y))
After this PR, they should be Prop-valued and take FunLike as a parameter:
/-- `MyHomClass F A B` states that `F` is a type of `MyClass.op`-preserving morphisms.
You should extend this class when you extend `MyHom`. -/
class MyHomClass (F : Type*) (A B : outParam <| Type*) [MyClass A] [MyClass B]
[FunLike F A B] : Prop :=
(map_op : ∀ (f : F) (x y : A), f (MyClass.op x y) = MyClass.op (f x) (f y))
(Note that A B stay marked as outParam even though they are not purely required to be so due to the FunLike parameter already filling them in. This is required to see through type synonyms, which is important in the category theory library. Also, I think keeping them as outParam is slightly faster.)
Similarly, MyEquivClass should take EquivLike as a parameter.
As a result, every mention of [MyHomClass F A B] should become [FunLike F A B] [MyHomClass F A B].
Remaining issues
Slower (failing) search
While overall this gives some great speedups, there are some cases that are noticeably slower. In particular, a failing application of a lemma such as map_mul is more expensive. This is due to suboptimal processing of arguments. For example:
variable [FunLike F M N] [Mul M] [Mul N] (f : F) (x : M) (y : M)
theorem map_mul [MulHomClass F M N] : f (x * y) = f x * f y
example [AddHomClass F A B] : f (x * y) = f x * f y := map_mul f _ _
Before this PR, applying map_mul f gives the goals [Mul ?M] [Mul ?N] [MulHomClass F ?M ?N]. Since M and N are out_params, [MulHomClass F ?M ?N] is synthesized first, supplies values for ?M and ?N and then the Mul M and Mul N instances can be found.
After this PR, the goals become [FunLike F ?M ?N] [Mul ?M] [Mul ?N] [MulHomClass F ?M ?N]. Now [FunLike F ?M ?N] is synthesized first, supplies values for ?M and ?N and then the Mul M and Mul N instances can be found, before trying MulHomClass F M N which fails. Since the Mul hierarchy is very big, this can be slow to fail, especially when there is no such Mul instance.
A long-term but harder to achieve solution would be to specify the order in which instance goals get solved. For example, we'd like to change the arguments to map_mul to look like [FunLike F M N] [Mul M] [Mul N] [highPriority <| MulHomClass F M N] because MulHomClass fails or succeeds much faster than the others.
As a consequence, the simpNF linter is much slower since by design it tries and fails to apply many map_ lemmas. The same issue occurs a few times in existing calls to simp [map_mul], where map_mul is tried "too soon" and fails. Thanks to the speedup of leanprover/lean4#2478 the impact is very limited, only in files that already were close to the timeout.
simp not firing sometimes
This affects map_smulₛₗ and related definitions. For simp lemmas Lean apparently uses a slightly different mechanism to find instances, so that rw can find every argument to map_smulₛₗ successfully but simp can't: leanprover/lean4#3701.
Missing instances due to unification failing
Especially in the category theory library, we might sometimes have a type A which is also accessible as a synonym (Bundled A hA).1. Instance synthesis doesn't always work if we have f : A →* B but x * y : (Bundled A hA).1 or vice versa. This seems to be mostly fixed by keeping A B as outParams in MulHomClass F A B. (Presumably because Lean will do a definitional check A =?= (Bundled A hA).1 instead of using the syntax in the discrimination tree.)
Workaround for issues
The timeouts can be worked around for now by specifying which map_mul we mean, either as map_mul f for some explicit f, or as e.g. MonoidHomClass.map_mul.
map_smulₛₗ not firing as simp lemma can be worked around by going back to the pre-FunLike situation and making LinearMap.map_smulₛₗ a simp lemma instead of the generic map_smulₛₗ. Writing simp [map_smulₛₗ _] also works.
Co-authored-by: Matthew Ballard <matt@mrb.email> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Scott Morrison <scott@tqft.net> Co-authored-by: Anne Baanen <Vierkantor@users.noreply.github.com>
Diff
@@ -1264,6 +1264,7 @@ theorem card_algHom_adjoin_integral (h : IsIntegral F α) (h_sep : (minpoly F α
simp only [adjoin.powerBasis_dim, adjoin.powerBasis_gen, minpoly_gen, h_sep, h_splits]
#align intermediate_field.card_alg_hom_adjoin_integral IntermediateField.card_algHom_adjoin_integral
+-- Apparently `K⟮root f⟯ →+* K⟮root f⟯` is expensive to unify during instance synthesis.
open FiniteDimensional AdjoinRoot in
/-- Let `f, g` be monic polynomials over `K`. If `f` is irreducible, and `g(x) - α` is irreducible
in `K⟮α⟯` with `α` a root of `f`, then `f(g(x))` is irreducible. -/
@@ -1278,7 +1279,8 @@ theorem _root_.Polynomial.irreducible_comp {f g : K[X]} (hfm : f.Monic) (hgm : g
· have := Fact.mk hf
intro e
apply not_irreducible_C ((g.map (algebraMap _ _)).coeff 0 - AdjoinSimple.gen K (root f))
- rw [map_sub, coeff_map, ← map_C, ← eq_C_of_natDegree_eq_zero e]
+ -- Needed to specialize `map_sub` to avoid a timeout #8386
+ rw [RingHom.map_sub, coeff_map, ← map_C, ← eq_C_of_natDegree_eq_zero e]
apply hg (AdjoinRoot f)
rw [AdjoinRoot.minpoly_root hf.ne_zero, hfm, inv_one, map_one, mul_one]
have H₁ : f.comp g ≠ 0 := fun h ↦ by simpa [hf', hg', natDegree_comp] using congr_arg natDegree h
feat({Ring|Field}Theory/Adjoin): add various results on adjoin (#9790)
Main changes:
- for intermediate fields: add
extendScalars_adjoin,restrictScalars_adjoin_of_algEquiv,sup_toSubalgebra_of_isAlgebraic,adjoin_toSubalgebra_of_isAlgebraic,adjoin_rank_le_of_isAlgebraic, and golfsup_toSubalgebra - for subalgebras: add
adjoin_eq_span_of_eq_span,adjoin_eq_span_basis,restrictScalars_adjoin,restrictScalars_adjoin_of_algEquiv,adjoin_rank_le
Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>
Diff
@@ -484,6 +484,31 @@ theorem restrictScalars_adjoin (K : IntermediateField F E) (S : Set E) :
restrictScalars F (adjoin K S) = adjoin F (K ∪ S) := by
rw [← adjoin_self _ K, adjoin_adjoin_left, adjoin_self _ K]
+variable {F} in
+theorem extendScalars_adjoin {K : IntermediateField F E} {S : Set E} (h : K ≤ adjoin F S) :
+ extendScalars h = adjoin K S := restrictScalars_injective F <| by
+ rw [extendScalars_restrictScalars, restrictScalars_adjoin]
+ exact le_antisymm (adjoin.mono F S _ <| Set.subset_union_right _ S) <| adjoin_le_iff.2 <|
+ Set.union_subset h (subset_adjoin F S)
+
+variable {F} in
+/-- If `E / L / F` and `E / L' / F` are two field extension towers, `L ≃ₐ[F] L'` is an isomorphism
+compatible with `E / L` and `E / L'`, then for any subset `S` of `E`, `L(S)` and `L'(S)` are
+equal as intermediate fields of `E / F`. -/
+theorem restrictScalars_adjoin_of_algEquiv
+ {L L' : Type*} [Field L] [Field L']
+ [Algebra F L] [Algebra L E] [Algebra F L'] [Algebra L' E]
+ [IsScalarTower F L E] [IsScalarTower F L' E] (i : L ≃ₐ[F] L')
+ (hi : algebraMap L E = (algebraMap L' E) ∘ i) (S : Set E) :
+ (adjoin L S).restrictScalars F = (adjoin L' S).restrictScalars F := by
+ apply_fun toSubfield using (fun K K' h ↦ by
+ ext x; change x ∈ K.toSubfield ↔ x ∈ K'.toSubfield; rw [h])
+ change Subfield.closure _ = Subfield.closure _
+ congr
+ ext x
+ exact ⟨fun ⟨y, h⟩ ↦ ⟨i y, by rw [← h, hi]; rfl⟩,
+ fun ⟨y, h⟩ ↦ ⟨i.symm y, by rw [← h, hi, Function.comp_apply, AlgEquiv.apply_symm_apply]⟩⟩
+
theorem algebra_adjoin_le_adjoin : Algebra.adjoin F S ≤ (adjoin F S).toSubalgebra :=
Algebra.adjoin_le (subset_adjoin _ _)
#align intermediate_field.algebra_adjoin_le_adjoin IntermediateField.algebra_adjoin_le_adjoin
@@ -648,37 +673,47 @@ theorem le_sup_toSubalgebra : E1.toSubalgebra ⊔ E2.toSubalgebra ≤ (E1 ⊔ E2
sup_le (show E1 ≤ E1 ⊔ E2 from le_sup_left) (show E2 ≤ E1 ⊔ E2 from le_sup_right)
#align intermediate_field.le_sup_to_subalgebra IntermediateField.le_sup_toSubalgebra
-theorem sup_toSubalgebra [h1 : FiniteDimensional K E1] [h2 : FiniteDimensional K E2] :
+theorem sup_toSubalgebra_of_isAlgebraic_right (halg : Algebra.IsAlgebraic K E2) :
(E1 ⊔ E2).toSubalgebra = E1.toSubalgebra ⊔ E2.toSubalgebra := by
- let S1 := E1.toSubalgebra
- let S2 := E2.toSubalgebra
- refine'
- le_antisymm
- (show _ ≤ (S1 ⊔ S2).toIntermediateField _ from
- sup_le (show S1 ≤ _ from le_sup_left) (show S2 ≤ _ from le_sup_right))
- (le_sup_toSubalgebra E1 E2)
- suffices IsField (S1 ⊔ S2 : Subalgebra K L) by
- intro x hx
- by_cases hx' : (⟨x, hx⟩ : (S1 ⊔ S2 : Subalgebra K L)) = 0
- · rw [← Subtype.coe_mk x, hx', Subalgebra.coe_zero, inv_zero]
- exact (S1 ⊔ S2).zero_mem
- · obtain ⟨y, h⟩ := this.mul_inv_cancel hx'
- exact (congr_arg (· ∈ S1 ⊔ S2) <| eq_inv_of_mul_eq_one_right <| Subtype.ext_iff.mp h).mp y.2
- exact
- isField_of_isIntegral_of_isField'
- (Algebra.isIntegral_sup.mpr
- ⟨Algebra.IsIntegral.of_finite K E1, Algebra.IsIntegral.of_finite K E2⟩)
- (Field.toIsField K)
-#align intermediate_field.sup_to_subalgebra IntermediateField.sup_toSubalgebra
-
-instance finiteDimensional_sup [h1 : FiniteDimensional K E1] [h2 : FiniteDimensional K E2] :
+ have : (adjoin E1 (E2 : Set L)).toSubalgebra = _ := adjoin_algebraic_toSubalgebra fun x h ↦
+ IsAlgebraic.tower_top E1 (isAlgebraic_iff.1 (halg ⟨x, h⟩))
+ apply_fun Subalgebra.restrictScalars K at this
+ erw [← restrictScalars_toSubalgebra, restrictScalars_adjoin,
+ Algebra.restrictScalars_adjoin] at this
+ exact this
+
+theorem sup_toSubalgebra_of_isAlgebraic_left (halg : Algebra.IsAlgebraic K E1) :
+ (E1 ⊔ E2).toSubalgebra = E1.toSubalgebra ⊔ E2.toSubalgebra := by
+ have := sup_toSubalgebra_of_isAlgebraic_right E2 E1 halg
+ rwa [sup_comm (a := E1), sup_comm (a := E1.toSubalgebra)]
+
+/-- The compositum of two intermediate fields is equal to the compositum of them
+as subalgebras, if one of them is algebraic over the base field. -/
+theorem sup_toSubalgebra_of_isAlgebraic
+ (halg : Algebra.IsAlgebraic K E1 ∨ Algebra.IsAlgebraic K E2) :
+ (E1 ⊔ E2).toSubalgebra = E1.toSubalgebra ⊔ E2.toSubalgebra :=
+ halg.elim (sup_toSubalgebra_of_isAlgebraic_left E1 E2)
+ (sup_toSubalgebra_of_isAlgebraic_right E1 E2)
+
+theorem sup_toSubalgebra_of_left [FiniteDimensional K E1] :
+ (E1 ⊔ E2).toSubalgebra = E1.toSubalgebra ⊔ E2.toSubalgebra :=
+ sup_toSubalgebra_of_isAlgebraic_left E1 E2 (Algebra.IsAlgebraic.of_finite K _)
+#align intermediate_field.sup_to_subalgebra IntermediateField.sup_toSubalgebra_of_left
+
+@[deprecated] alias sup_toSubalgebra := sup_toSubalgebra_of_left
+
+theorem sup_toSubalgebra_of_right [FiniteDimensional K E2] :
+ (E1 ⊔ E2).toSubalgebra = E1.toSubalgebra ⊔ E2.toSubalgebra :=
+ sup_toSubalgebra_of_isAlgebraic_right E1 E2 (Algebra.IsAlgebraic.of_finite K _)
+
+instance finiteDimensional_sup [FiniteDimensional K E1] [FiniteDimensional K E2] :
FiniteDimensional K (E1 ⊔ E2 : IntermediateField K L) := by
let g := Algebra.TensorProduct.productMap E1.val E2.val
suffices g.range = (E1 ⊔ E2).toSubalgebra by
have h : FiniteDimensional K (Subalgebra.toSubmodule g.range) :=
g.toLinearMap.finiteDimensional_range
rwa [this] at h
- rw [Algebra.TensorProduct.productMap_range, E1.range_val, E2.range_val, sup_toSubalgebra]
+ rw [Algebra.TensorProduct.productMap_range, E1.range_val, E2.range_val, sup_toSubalgebra_of_left]
#align intermediate_field.finite_dimensional_sup IntermediateField.finiteDimensional_sup
variable {ι : Type*} {t : ι → IntermediateField K L}
@@ -749,6 +784,61 @@ theorem isSplittingField_iSup {p : ι → K[X]}
end Supremum
+section Tower
+
+variable (E)
+
+variable {K : Type*} [Field K] [Algebra F K] [Algebra E K] [IsScalarTower F E K]
+
+/-- If `K / E / F` is a field extension tower, `L` is an intermediate field of `K / F`, such that
+either `E / F` or `L / F` is algebraic, then `E(L) = E[L]`. -/
+theorem adjoin_toSubalgebra_of_isAlgebraic (L : IntermediateField F K)
+ (halg : Algebra.IsAlgebraic F E ∨ Algebra.IsAlgebraic F L) :
+ (adjoin E (L : Set K)).toSubalgebra = Algebra.adjoin E (L : Set K) := by
+ let i := IsScalarTower.toAlgHom F E K
+ let E' := i.fieldRange
+ let i' : E ≃ₐ[F] E' := AlgEquiv.ofInjectiveField i
+ have hi : algebraMap E K = (algebraMap E' K) ∘ i' := by ext x; rfl
+ apply_fun _ using Subalgebra.restrictScalars_injective F
+ erw [← restrictScalars_toSubalgebra, restrictScalars_adjoin_of_algEquiv i' hi,
+ Algebra.restrictScalars_adjoin_of_algEquiv i' hi, restrictScalars_adjoin,
+ Algebra.restrictScalars_adjoin]
+ exact E'.sup_toSubalgebra_of_isAlgebraic L (halg.imp i'.isAlgebraic id)
+
+theorem adjoin_toSubalgebra_of_isAlgebraic_left (L : IntermediateField F K)
+ (halg : Algebra.IsAlgebraic F E) :
+ (adjoin E (L : Set K)).toSubalgebra = Algebra.adjoin E (L : Set K) :=
+ adjoin_toSubalgebra_of_isAlgebraic E L (Or.inl halg)
+
+theorem adjoin_toSubalgebra_of_isAlgebraic_right (L : IntermediateField F K)
+ (halg : Algebra.IsAlgebraic F L) :
+ (adjoin E (L : Set K)).toSubalgebra = Algebra.adjoin E (L : Set K) :=
+ adjoin_toSubalgebra_of_isAlgebraic E L (Or.inr halg)
+
+/-- If `K / E / F` is a field extension tower, `L` is an intermediate field of `K / F`, such that
+either `E / F` or `L / F` is algebraic, then `[E(L) : E] ≤ [L : F]`. A corollary of
+`Subalgebra.adjoin_rank_le` since in this case `E(L) = E[L]`. -/
+theorem adjoin_rank_le_of_isAlgebraic (L : IntermediateField F K)
+ (halg : Algebra.IsAlgebraic F E ∨ Algebra.IsAlgebraic F L) :
+ Module.rank E (adjoin E (L : Set K)) ≤ Module.rank F L := by
+ have h : (adjoin E (L.toSubalgebra : Set K)).toSubalgebra =
+ Algebra.adjoin E (L.toSubalgebra : Set K) :=
+ L.adjoin_toSubalgebra_of_isAlgebraic E halg
+ have := L.toSubalgebra.adjoin_rank_le E
+ rwa [(Subalgebra.equivOfEq _ _ h).symm.toLinearEquiv.rank_eq] at this
+
+theorem adjoin_rank_le_of_isAlgebraic_left (L : IntermediateField F K)
+ (halg : Algebra.IsAlgebraic F E) :
+ Module.rank E (adjoin E (L : Set K)) ≤ Module.rank F L :=
+ adjoin_rank_le_of_isAlgebraic E L (Or.inl halg)
+
+theorem adjoin_rank_le_of_isAlgebraic_right (L : IntermediateField F K)
+ (halg : Algebra.IsAlgebraic F L) :
+ Module.rank E (adjoin E (L : Set K)) ≤ Module.rank F L :=
+ adjoin_rank_le_of_isAlgebraic E L (Or.inr halg)
+
+end Tower
+
open Set CompleteLattice
/- Porting note: this was tagged `simp`, but the LHS can be simplified now that the notation
feat: add equivMap[OfInjective] for subalgebra and intermediate field (#9709)
... parallel to Subring.equivMapOfInjective, Subsemiring.equivMapOfInjective, etc. etc.
Diff
@@ -315,6 +315,24 @@ theorem _root_.AlgEquiv.fieldRange_eq_top (f : E ≃ₐ[F] K) :
end Lattice
+section equivMap
+
+variable {F : Type*} [Field F] {E : Type*} [Field E] [Algebra F E]
+ {K : Type*} [Field K] [Algebra F K] (L : IntermediateField F E) (f : E →ₐ[F] K)
+
+theorem fieldRange_comp_val : (f.comp L.val).fieldRange = L.map f := toSubalgebra_injective <| by
+ rw [toSubalgebra_map, AlgHom.fieldRange_toSubalgebra, AlgHom.range_comp, range_val]
+
+/-- An intermediate field is isomorphic to its image under an `AlgHom`
+(which is automatically injective) -/
+noncomputable def equivMap : L ≃ₐ[F] L.map f :=
+ (AlgEquiv.ofInjective _ (f.comp L.val).injective).trans (equivOfEq (fieldRange_comp_val L f))
+
+@[simp]
+theorem coe_equivMap_apply (x : L) : ↑(equivMap L f x) = f x := rfl
+
+end equivMap
+
section AdjoinDef
variable (F : Type*) [Field F] {E : Type*} [Field E] [Algebra F E] (S : Set E)
Diff
@@ -1265,213 +1265,12 @@ theorem induction_on_adjoin [FiniteDimensional F E] (P : IntermediateField F E
end Induction
-section AlgHomMkAdjoinSplits
-
-variable (F E K : Type*) [Field F] [Field E] [Field K] [Algebra F E] [Algebra F K] {S : Set E}
-
-/-- Lifts `L → K` of `F → K` -/
-structure Lifts where
- /-- The domain of a lift. -/
- carrier : IntermediateField F E
- /-- The lifted RingHom, expressed as an AlgHom. -/
- emb : carrier →ₐ[F] K
-#align intermediate_field.lifts IntermediateField.Lifts
-
-variable {F E K}
-
-instance : PartialOrder (Lifts F E K) where
- le L₁ L₂ := ∃ h : L₁.carrier ≤ L₂.carrier, ∀ x, L₂.emb (inclusion h x) = L₁.emb x
- le_refl L := ⟨le_rfl, by simp⟩
- le_trans L₁ L₂ L₃ := by
- rintro ⟨h₁₂, h₁₂'⟩ ⟨h₂₃, h₂₃'⟩
- refine ⟨h₁₂.trans h₂₃, fun _ ↦ ?_⟩
- rw [← inclusion_inclusion h₁₂ h₂₃, h₂₃', h₁₂']
- le_antisymm := by
- rintro ⟨L₁, e₁⟩ ⟨L₂, e₂⟩ ⟨h₁₂, h₁₂'⟩ ⟨h₂₁, h₂₁'⟩
- obtain rfl : L₁ = L₂ := h₁₂.antisymm h₂₁
- congr
- exact AlgHom.ext h₂₁'
-
-noncomputable instance : OrderBot (Lifts F E K) where
- bot := ⟨⊥, (Algebra.ofId F K).comp (botEquiv F E)⟩
- bot_le L := ⟨bot_le, fun x ↦ by
- obtain ⟨x, rfl⟩ := (botEquiv F E).symm.surjective x
- simp_rw [AlgHom.comp_apply, AlgHom.coe_coe, AlgEquiv.apply_symm_apply]
- exact L.emb.commutes x⟩
-
-noncomputable instance : Inhabited (Lifts F E K) :=
- ⟨⊥⟩
-
-/-- A chain of lifts has an upper bound. -/
-theorem Lifts.exists_upper_bound (c : Set (Lifts F E K)) (hc : IsChain (· ≤ ·) c) :
- ∃ ub, ∀ a ∈ c, a ≤ ub := by
- let t (i : ↑(insert ⊥ c)) := i.val.carrier
- let t' (i) := (t i).toSubalgebra
- have hc := hc.insert fun _ _ _ ↦ .inl bot_le
- have dir : Directed (· ≤ ·) t := hc.directedOn.directed_val.mono_comp _ fun _ _ h ↦ h.1
- refine ⟨⟨iSup t, (Subalgebra.iSupLift t' dir (fun i ↦ i.val.emb) (fun i j h ↦ ?_) _ rfl).comp
- (Subalgebra.equivOfEq _ _ <| toSubalgebra_iSup_of_directed dir)⟩,
- fun L hL ↦ have hL := Set.mem_insert_of_mem ⊥ hL; ⟨le_iSup t ⟨L, hL⟩, fun x ↦ ?_⟩⟩
- · refine AlgHom.ext fun x ↦ (hc.total i.2 j.2).elim (fun hij ↦ (hij.snd x).symm) fun hji ↦ ?_
- erw [AlgHom.comp_apply, ← hji.snd (Subalgebra.inclusion h x),
- inclusion_inclusion, inclusion_self, AlgHom.id_apply x]
- · dsimp only [AlgHom.comp_apply]
- exact Subalgebra.iSupLift_inclusion (K := t') (i := ⟨L, hL⟩) x (le_iSup t' ⟨L, hL⟩)
-#align intermediate_field.lifts.exists_upper_bound IntermediateField.Lifts.exists_upper_bound
-
-/-- Given a lift `x` and an integral element `s : E` over `x.carrier` whose conjugates over
-`x.carrier` are all in `K`, we can extend the lift to a lift whose carrier contains `s`. -/
-theorem Lifts.exists_lift_of_splits' (x : Lifts F E K) {s : E} (h1 : IsIntegral x.carrier s)
- (h2 : (minpoly x.carrier s).Splits x.emb.toRingHom) : ∃ y, x ≤ y ∧ s ∈ y.carrier :=
- have I2 := (minpoly.degree_pos h1).ne'
- letI : Algebra x.carrier K := x.emb.toRingHom.toAlgebra
- let carrier := x.carrier⟮s⟯.restrictScalars F
- letI : Algebra x.carrier carrier := x.carrier⟮s⟯.toSubalgebra.algebra
- let φ : carrier →ₐ[x.carrier] K := ((algHomAdjoinIntegralEquiv x.carrier h1).symm
- ⟨rootOfSplits x.emb.toRingHom h2 I2, by
- rw [mem_aroots, and_iff_right (minpoly.ne_zero h1)]
- exact map_rootOfSplits x.emb.toRingHom h2 I2⟩)
- ⟨⟨carrier, (@algHomEquivSigma F x.carrier carrier K _ _ _ _ _ _ _ _
- (IsScalarTower.of_algebraMap_eq fun _ ↦ rfl)).symm ⟨x.emb, φ⟩⟩,
- ⟨fun z hz ↦ algebraMap_mem x.carrier⟮s⟯ ⟨z, hz⟩, φ.commutes⟩,
- mem_adjoin_simple_self x.carrier s⟩
-
-/-- Given an integral element `s : E` over `F` whose `F`-conjugates are all in `K`,
-any lift can be extended to one whose carrier contains `s`. -/
-theorem Lifts.exists_lift_of_splits (x : Lifts F E K) {s : E} (h1 : IsIntegral F s)
- (h2 : (minpoly F s).Splits (algebraMap F K)) : ∃ y, x ≤ y ∧ s ∈ y.carrier :=
- Lifts.exists_lift_of_splits' x h1.tower_top <| h1.minpoly_splits_tower_top' <| by
- rwa [← x.emb.comp_algebraMap] at h2
-#align intermediate_field.lifts.exists_lift_of_splits IntermediateField.Lifts.exists_lift_of_splits
-
-section
-
-private theorem exists_algHom_adjoin_of_splits'' {L : IntermediateField F E}
- (f : L →ₐ[F] K) (hK : ∀ s ∈ S, IsIntegral L s ∧ (minpoly L s).Splits f.toRingHom) :
- ∃ φ : adjoin L S →ₐ[F] K, φ.comp (IsScalarTower.toAlgHom F L _) = f := by
- obtain ⟨φ, hfφ, hφ⟩ := zorn_nonempty_Ici₀ _
- (fun c _ hc _ _ ↦ Lifts.exists_upper_bound c hc) ⟨L, f⟩ le_rfl
- refine ⟨φ.emb.comp (inclusion <| (le_extendScalars_iff hfφ.1 <| adjoin L S).mp <|
- adjoin_le_iff.mpr fun s h ↦ ?_), AlgHom.ext hfφ.2⟩
- letI := (inclusion hfφ.1).toAlgebra
- letI : SMul L φ.carrier := Algebra.toSMul
- have : IsScalarTower L φ.carrier E := ⟨(smul_assoc · (· : E))⟩
- have := φ.exists_lift_of_splits' (hK s h).1.tower_top ((hK s h).1.minpoly_splits_tower_top' ?_)
- · obtain ⟨y, h1, h2⟩ := this; exact (hφ y h1).1 h2
- · convert (hK s h).2; ext; apply hfφ.2
-
-variable {L : Type*} [Field L] [Algebra F L] [Algebra L E] [IsScalarTower F L E]
- (f : L →ₐ[F] K) (hK : ∀ s ∈ S, IsIntegral L s ∧ (minpoly L s).Splits f.toRingHom)
-
-theorem exists_algHom_adjoin_of_splits' :
- ∃ φ : adjoin L S →ₐ[F] K, φ.comp (IsScalarTower.toAlgHom F L _) = f := by
- let L' := (IsScalarTower.toAlgHom F L E).fieldRange
- let f' : L' →ₐ[F] K := f.comp (AlgEquiv.ofInjectiveField _).symm.toAlgHom
- have := exists_algHom_adjoin_of_splits'' f' (S := S) fun s hs ↦ ?_
- · obtain ⟨φ, hφ⟩ := this; refine ⟨φ.comp <|
- inclusion (?_ : (adjoin L S).restrictScalars F ≤ (adjoin L' S).restrictScalars F), ?_⟩
- · simp_rw [← SetLike.coe_subset_coe, coe_restrictScalars, adjoin_subset_adjoin_iff]
- refine ⟨subset_adjoin_of_subset_left S (F := L'.toSubfield) le_rfl, subset_adjoin _ _⟩
- · ext x; exact congr($hφ _).trans (congr_arg f <| AlgEquiv.symm_apply_apply _ _)
- letI : Algebra L L' := (AlgEquiv.ofInjectiveField _).toRingEquiv.toRingHom.toAlgebra
- have : IsScalarTower L L' E := IsScalarTower.of_algebraMap_eq' rfl
- refine ⟨(hK s hs).1.tower_top, (hK s hs).1.minpoly_splits_tower_top' ?_⟩
- convert (hK s hs).2; ext; exact congr_arg f (AlgEquiv.symm_apply_apply _ _)
-
-theorem exists_algHom_of_adjoin_splits' (hS : adjoin L S = ⊤) :
- ∃ φ : E →ₐ[F] K, φ.comp (IsScalarTower.toAlgHom F L E) = f :=
- have ⟨φ, hφ⟩ := exists_algHom_adjoin_of_splits' f hK
- ⟨φ.comp (((equivOfEq hS).trans topEquiv).symm.toAlgHom.restrictScalars F), hφ⟩
-
-theorem exists_algHom_of_splits' (hK : ∀ s : E, IsIntegral L s ∧ (minpoly L s).Splits f.toRingHom) :
- ∃ φ : E →ₐ[F] K, φ.comp (IsScalarTower.toAlgHom F L E) = f :=
- exists_algHom_of_adjoin_splits' f (fun x _ ↦ hK x) (adjoin_univ L E)
-
-end
-
-variable (hK : ∀ s ∈ S, IsIntegral F s ∧ (minpoly F s).Splits (algebraMap F K))
- (hK' : ∀ s : E, IsIntegral F s ∧ (minpoly F s).Splits (algebraMap F K))
- {L : IntermediateField F E} (f : L →ₐ[F] K) (hL : L ≤ adjoin F S)
-
--- The following uses the hypothesis `hK`.
-
-theorem exists_algHom_adjoin_of_splits : ∃ φ : adjoin F S →ₐ[F] K, φ.comp (inclusion hL) = f := by
- obtain ⟨φ, hfφ, hφ⟩ := zorn_nonempty_Ici₀ _
- (fun c _ hc _ _ ↦ Lifts.exists_upper_bound c hc) ⟨L, f⟩ le_rfl
- refine ⟨φ.emb.comp (inclusion <| adjoin_le_iff.mpr fun s hs ↦ ?_), ?_⟩
- · rcases φ.exists_lift_of_splits (hK s hs).1 (hK s hs).2 with ⟨y, h1, h2⟩
- exact (hφ y h1).1 h2
- · ext; apply hfφ.2
-
-theorem nonempty_algHom_adjoin_of_splits : Nonempty (adjoin F S →ₐ[F] K) :=
- have ⟨φ, _⟩ := exists_algHom_adjoin_of_splits hK (⊥ : Lifts F E K).emb bot_le; ⟨φ⟩
-#align intermediate_field.alg_hom_mk_adjoin_splits IntermediateField.nonempty_algHom_adjoin_of_splits
-
-variable (hS : adjoin F S = ⊤)
-
-theorem exists_algHom_of_adjoin_splits : ∃ φ : E →ₐ[F] K, φ.comp L.val = f :=
- have ⟨φ, hφ⟩ := exists_algHom_adjoin_of_splits hK f (hS.symm ▸ le_top)
- ⟨φ.comp ((equivOfEq hS).trans topEquiv).symm.toAlgHom, hφ⟩
-
-theorem nonempty_algHom_of_adjoin_splits : Nonempty (E →ₐ[F] K) :=
- have ⟨φ, _⟩ := exists_algHom_of_adjoin_splits hK (⊥ : Lifts F E K).emb hS; ⟨φ⟩
-#align intermediate_field.alg_hom_mk_adjoin_splits' IntermediateField.nonempty_algHom_of_adjoin_splits
-
-variable {x : E} (hx : x ∈ adjoin F S) {y : K} (hy : aeval y (minpoly F x) = 0)
-
-theorem exists_algHom_adjoin_of_splits_of_aeval : ∃ φ : adjoin F S →ₐ[F] K, φ ⟨x, hx⟩ = y := by
- have ix := isAlgebraic_adjoin (fun s hs ↦ (hK s hs).1) ⟨x, hx⟩
- rw [isAlgebraic_iff_isIntegral, isIntegral_iff] at ix
- obtain ⟨φ, hφ⟩ := exists_algHom_adjoin_of_splits hK ((algHomAdjoinIntegralEquiv F ix).symm
- ⟨y, mem_aroots.mpr ⟨minpoly.ne_zero ix, hy⟩⟩) (adjoin_simple_le_iff.mpr hx)
- exact ⟨φ, (FunLike.congr_fun hφ <| AdjoinSimple.gen F x).trans <|
- algHomAdjoinIntegralEquiv_symm_apply_gen F ix _⟩
-
-theorem exists_algHom_of_adjoin_splits_of_aeval : ∃ φ : E →ₐ[F] K, φ x = y :=
- have ⟨φ, hφ⟩ := exists_algHom_adjoin_of_splits_of_aeval hK (hS ▸ mem_top) hy
- ⟨φ.comp ((equivOfEq hS).trans topEquiv).symm.toAlgHom, hφ⟩
-
-/- The following uses the hypothesis
- (hK' : ∀ s : E, IsIntegral F s ∧ (minpoly F s).Splits (algebraMap F K)) -/
-
-theorem exists_algHom_of_splits : ∃ φ : E →ₐ[F] K, φ.comp L.val = f :=
- exists_algHom_of_adjoin_splits (fun x _ ↦ hK' x) f (adjoin_univ F E)
-
-theorem nonempty_algHom_of_splits : Nonempty (E →ₐ[F] K) :=
- nonempty_algHom_of_adjoin_splits (fun x _ ↦ hK' x) (adjoin_univ F E)
-
-theorem exists_algHom_of_splits_of_aeval : ∃ φ : E →ₐ[F] K, φ x = y :=
- exists_algHom_of_adjoin_splits_of_aeval (fun x _ ↦ hK' x) (adjoin_univ F E) hy
-
-end AlgHomMkAdjoinSplits
-
end IntermediateField
-section Algebra.IsAlgebraic
-
-/-- Let `K/F` be an algebraic extension of fields and `L` a field in which all the minimal
-polynomial over `F` of elements of `K` splits. Then, for `x ∈ K`, the images of `x` by the
-`F`-algebra morphisms from `K` to `L` are exactly the roots in `L` of the minimal polynomial
-of `x` over `F`. -/
-theorem Algebra.IsAlgebraic.range_eval_eq_rootSet_minpoly_of_splits {F K : Type*} (L : Type*)
- [Field F] [Field K] [Field L] [Algebra F L] [Algebra F K]
- (hA : ∀ x : K, (minpoly F x).Splits (algebraMap F L))
- (hK : Algebra.IsAlgebraic F K) (x : K) :
- (Set.range fun (ψ : K →ₐ[F] L) => ψ x) = (minpoly F x).rootSet L := by
- ext a
- rw [mem_rootSet_of_ne (minpoly.ne_zero (hK.isIntegral x))]
- refine ⟨fun ⟨ψ, hψ⟩ ↦ ?_, fun ha ↦ IntermediateField.exists_algHom_of_splits_of_aeval
- (fun x ↦ ⟨hK.isIntegral x, hA x⟩) ha⟩
- rw [← hψ, Polynomial.aeval_algHom_apply ψ x, minpoly.aeval, map_zero]
-
-end Algebra.IsAlgebraic
-
-section PowerBasis
+namespace PowerBasis
variable {K L : Type*} [Field K] [Field L] [Algebra K L]
-namespace PowerBasis
-
open IntermediateField
/-- `pb.equivAdjoinSimple` is the equivalence between `K⟮pb.gen⟯` and `L` itself. -/
@@ -1505,5 +1304,3 @@ theorem equivAdjoinSimple_symm_gen (pb : PowerBasis K L) :
#align power_basis.equiv_adjoin_simple_symm_gen PowerBasis.equivAdjoinSimple_symm_gen
end PowerBasis
-
-end PowerBasis
feat (FieldTheory/Adjoin): generalize IntermediateField.exists_algHom_of_splits (#9392)
-
Extract from
Lifts.exists_lift_of_splitsa version that assumes integrality and splitting of minimal polynomial over an intermediate field (either actual IntermediateField or middle of IsScalarTower) rather than the base field -
Generalize
exists_algHom_adjoin_of_splits,exists_algHom_of_adjoin_splits,exists_algHom_of_splitsto the same setting -
Prove the extension lemma
surjective_comp_algebraMap_of_isAlgebraic/isSeparableto be used in the proof that an epimorphism in the category of fields is a purely inseparable extension.
Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>
Diff
@@ -1319,28 +1319,76 @@ theorem Lifts.exists_upper_bound (c : Set (Lifts F E K)) (hc : IsChain (· ≤
exact Subalgebra.iSupLift_inclusion (K := t') (i := ⟨L, hL⟩) x (le_iSup t' ⟨L, hL⟩)
#align intermediate_field.lifts.exists_upper_bound IntermediateField.Lifts.exists_upper_bound
-/-- Given an element `s : E` whose conjugates are all in `K`, any lift can be extended to one
- whose carrier contains `s`. -/
-theorem Lifts.exists_lift_of_splits (x : Lifts F E K) {s : E} (h1 : IsIntegral F s)
- (h2 : (minpoly F s).Splits (algebraMap F K)) : ∃ y, x ≤ y ∧ s ∈ y.carrier :=
- have I1 : IsIntegral x.carrier s := h1.tower_top
- have I2 := (minpoly.degree_pos I1).ne'
- have key : (minpoly x.carrier s).Splits x.emb.toRingHom :=
- splits_of_splits_of_dvd _ (map_ne_zero (minpoly.ne_zero h1))
- ((splits_map_iff _ _).2 (x.emb.comp_algebraMap ▸ h2)) (minpoly.dvd_map_of_isScalarTower _ _ _)
+/-- Given a lift `x` and an integral element `s : E` over `x.carrier` whose conjugates over
+`x.carrier` are all in `K`, we can extend the lift to a lift whose carrier contains `s`. -/
+theorem Lifts.exists_lift_of_splits' (x : Lifts F E K) {s : E} (h1 : IsIntegral x.carrier s)
+ (h2 : (minpoly x.carrier s).Splits x.emb.toRingHom) : ∃ y, x ≤ y ∧ s ∈ y.carrier :=
+ have I2 := (minpoly.degree_pos h1).ne'
letI : Algebra x.carrier K := x.emb.toRingHom.toAlgebra
let carrier := x.carrier⟮s⟯.restrictScalars F
letI : Algebra x.carrier carrier := x.carrier⟮s⟯.toSubalgebra.algebra
- let φ : carrier →ₐ[x.carrier] K := ((algHomAdjoinIntegralEquiv x.carrier I1).symm
- ⟨rootOfSplits x.emb.toRingHom key I2, by
- rw [mem_aroots, and_iff_right (minpoly.ne_zero I1)]
- exact map_rootOfSplits x.emb.toRingHom key I2⟩)
+ let φ : carrier →ₐ[x.carrier] K := ((algHomAdjoinIntegralEquiv x.carrier h1).symm
+ ⟨rootOfSplits x.emb.toRingHom h2 I2, by
+ rw [mem_aroots, and_iff_right (minpoly.ne_zero h1)]
+ exact map_rootOfSplits x.emb.toRingHom h2 I2⟩)
⟨⟨carrier, (@algHomEquivSigma F x.carrier carrier K _ _ _ _ _ _ _ _
(IsScalarTower.of_algebraMap_eq fun _ ↦ rfl)).symm ⟨x.emb, φ⟩⟩,
⟨fun z hz ↦ algebraMap_mem x.carrier⟮s⟯ ⟨z, hz⟩, φ.commutes⟩,
mem_adjoin_simple_self x.carrier s⟩
+
+/-- Given an integral element `s : E` over `F` whose `F`-conjugates are all in `K`,
+any lift can be extended to one whose carrier contains `s`. -/
+theorem Lifts.exists_lift_of_splits (x : Lifts F E K) {s : E} (h1 : IsIntegral F s)
+ (h2 : (minpoly F s).Splits (algebraMap F K)) : ∃ y, x ≤ y ∧ s ∈ y.carrier :=
+ Lifts.exists_lift_of_splits' x h1.tower_top <| h1.minpoly_splits_tower_top' <| by
+ rwa [← x.emb.comp_algebraMap] at h2
#align intermediate_field.lifts.exists_lift_of_splits IntermediateField.Lifts.exists_lift_of_splits
+section
+
+private theorem exists_algHom_adjoin_of_splits'' {L : IntermediateField F E}
+ (f : L →ₐ[F] K) (hK : ∀ s ∈ S, IsIntegral L s ∧ (minpoly L s).Splits f.toRingHom) :
+ ∃ φ : adjoin L S →ₐ[F] K, φ.comp (IsScalarTower.toAlgHom F L _) = f := by
+ obtain ⟨φ, hfφ, hφ⟩ := zorn_nonempty_Ici₀ _
+ (fun c _ hc _ _ ↦ Lifts.exists_upper_bound c hc) ⟨L, f⟩ le_rfl
+ refine ⟨φ.emb.comp (inclusion <| (le_extendScalars_iff hfφ.1 <| adjoin L S).mp <|
+ adjoin_le_iff.mpr fun s h ↦ ?_), AlgHom.ext hfφ.2⟩
+ letI := (inclusion hfφ.1).toAlgebra
+ letI : SMul L φ.carrier := Algebra.toSMul
+ have : IsScalarTower L φ.carrier E := ⟨(smul_assoc · (· : E))⟩
+ have := φ.exists_lift_of_splits' (hK s h).1.tower_top ((hK s h).1.minpoly_splits_tower_top' ?_)
+ · obtain ⟨y, h1, h2⟩ := this; exact (hφ y h1).1 h2
+ · convert (hK s h).2; ext; apply hfφ.2
+
+variable {L : Type*} [Field L] [Algebra F L] [Algebra L E] [IsScalarTower F L E]
+ (f : L →ₐ[F] K) (hK : ∀ s ∈ S, IsIntegral L s ∧ (minpoly L s).Splits f.toRingHom)
+
+theorem exists_algHom_adjoin_of_splits' :
+ ∃ φ : adjoin L S →ₐ[F] K, φ.comp (IsScalarTower.toAlgHom F L _) = f := by
+ let L' := (IsScalarTower.toAlgHom F L E).fieldRange
+ let f' : L' →ₐ[F] K := f.comp (AlgEquiv.ofInjectiveField _).symm.toAlgHom
+ have := exists_algHom_adjoin_of_splits'' f' (S := S) fun s hs ↦ ?_
+ · obtain ⟨φ, hφ⟩ := this; refine ⟨φ.comp <|
+ inclusion (?_ : (adjoin L S).restrictScalars F ≤ (adjoin L' S).restrictScalars F), ?_⟩
+ · simp_rw [← SetLike.coe_subset_coe, coe_restrictScalars, adjoin_subset_adjoin_iff]
+ refine ⟨subset_adjoin_of_subset_left S (F := L'.toSubfield) le_rfl, subset_adjoin _ _⟩
+ · ext x; exact congr($hφ _).trans (congr_arg f <| AlgEquiv.symm_apply_apply _ _)
+ letI : Algebra L L' := (AlgEquiv.ofInjectiveField _).toRingEquiv.toRingHom.toAlgebra
+ have : IsScalarTower L L' E := IsScalarTower.of_algebraMap_eq' rfl
+ refine ⟨(hK s hs).1.tower_top, (hK s hs).1.minpoly_splits_tower_top' ?_⟩
+ convert (hK s hs).2; ext; exact congr_arg f (AlgEquiv.symm_apply_apply _ _)
+
+theorem exists_algHom_of_adjoin_splits' (hS : adjoin L S = ⊤) :
+ ∃ φ : E →ₐ[F] K, φ.comp (IsScalarTower.toAlgHom F L E) = f :=
+ have ⟨φ, hφ⟩ := exists_algHom_adjoin_of_splits' f hK
+ ⟨φ.comp (((equivOfEq hS).trans topEquiv).symm.toAlgHom.restrictScalars F), hφ⟩
+
+theorem exists_algHom_of_splits' (hK : ∀ s : E, IsIntegral L s ∧ (minpoly L s).Splits f.toRingHom) :
+ ∃ φ : E →ₐ[F] K, φ.comp (IsScalarTower.toAlgHom F L E) = f :=
+ exists_algHom_of_adjoin_splits' f (fun x _ ↦ hK x) (adjoin_univ L E)
+
+end
+
variable (hK : ∀ s ∈ S, IsIntegral F s ∧ (minpoly F s).Splits (algebraMap F K))
(hK' : ∀ s : E, IsIntegral F s ∧ (minpoly F s).Splits (algebraMap F K))
{L : IntermediateField F E} (f : L →ₐ[F] K) (hL : L ≤ adjoin F S)
Diff
@@ -799,6 +799,15 @@ theorem exists_finset_of_mem_supr'' {ι : Type*} {f : ι → IntermediateField F
exact minpoly.ne_zero (isIntegral_iff.mp (h i ⟨x1, hx1⟩).isIntegral)
#align intermediate_field.exists_finset_of_mem_supr'' IntermediateField.exists_finset_of_mem_supr''
+theorem exists_finset_of_mem_adjoin {S : Set E} {x : E} (hx : x ∈ adjoin F S) :
+ ∃ T : Finset E, (T : Set E) ⊆ S ∧ x ∈ adjoin F (T : Set E) := by
+ simp_rw [← biSup_adjoin_simple S, ← iSup_subtype''] at hx
+ obtain ⟨s, hx'⟩ := exists_finset_of_mem_iSup hx
+ refine ⟨s.image Subtype.val, by simp, SetLike.le_def.mp ?_ hx'⟩
+ simp_rw [Finset.coe_image, iSup_le_iff, adjoin_le_iff]
+ rintro _ h _ rfl
+ exact subset_adjoin F _ ⟨_, h, rfl⟩
+
end AdjoinSimple
end AdjoinDef
Diff
@@ -558,6 +558,9 @@ def AdjoinSimple.gen : F⟮α⟯ :=
#align intermediate_field.adjoin_simple.gen IntermediateField.AdjoinSimple.gen
@[simp]
+theorem AdjoinSimple.coe_gen : (AdjoinSimple.gen F α : E) = α :=
+ rfl
+
theorem AdjoinSimple.algebraMap_gen : algebraMap F⟮α⟯ E (AdjoinSimple.gen F α) = α :=
rfl
#align intermediate_field.adjoin_simple.algebra_map_gen IntermediateField.AdjoinSimple.algebraMap_gen
@@ -866,6 +869,12 @@ theorem finrank_bot : finrank F (⊥ : IntermediateField F E) = 1 := by rw [finr
@[simp] protected theorem finrank_top : finrank (⊤ : IntermediateField F E) E = 1 :=
rank_eq_one_iff_finrank_eq_one.mp IntermediateField.rank_top
+@[simp] theorem rank_top' : Module.rank F (⊤ : IntermediateField F E) = Module.rank F E :=
+ rank_top F E
+
+@[simp] theorem finrank_top' : finrank F (⊤ : IntermediateField F E) = finrank F E :=
+ finrank_top F E
+
theorem rank_adjoin_eq_one_iff : Module.rank F (adjoin F S) = 1 ↔ S ⊆ (⊥ : IntermediateField F E) :=
Iff.trans rank_eq_one_iff adjoin_eq_bot_iff
#align intermediate_field.rank_adjoin_eq_one_iff IntermediateField.rank_adjoin_eq_one_iff
@@ -926,9 +935,11 @@ end AdjoinIntermediateFieldLattice
section AdjoinIntegralElement
+universe u
+
variable (F : Type*) [Field F] {E : Type*} [Field E] [Algebra F E] {α : E}
-variable {K : Type*} [Field K] [Algebra F K]
+variable {K : Type u} [Field K] [Algebra F K]
theorem minpoly_gen (α : E) :
minpoly F (AdjoinSimple.gen F α) = minpoly F α := by
@@ -974,6 +985,9 @@ theorem adjoinRootEquivAdjoin_apply_root (h : IsIntegral F α) :
AdjoinRoot.lift_root (aeval_gen_minpoly F α)
#align intermediate_field.adjoin_root_equiv_adjoin_apply_root IntermediateField.adjoinRootEquivAdjoin_apply_root
+theorem adjoin_root_eq_top (p : K[X]) [Fact (Irreducible p)] : K⟮AdjoinRoot.root p⟯ = ⊤ :=
+ (eq_adjoin_of_eq_algebra_adjoin K _ ⊤ (AdjoinRoot.adjoinRoot_eq_top (f := p)).symm).symm
+
section PowerBasis
variable {L : Type*} [Field L] [Algebra K L]
@@ -1133,6 +1147,56 @@ theorem card_algHom_adjoin_integral (h : IsIntegral F α) (h_sep : (minpoly F α
simp only [adjoin.powerBasis_dim, adjoin.powerBasis_gen, minpoly_gen, h_sep, h_splits]
#align intermediate_field.card_alg_hom_adjoin_integral IntermediateField.card_algHom_adjoin_integral
+open FiniteDimensional AdjoinRoot in
+/-- Let `f, g` be monic polynomials over `K`. If `f` is irreducible, and `g(x) - α` is irreducible
+in `K⟮α⟯` with `α` a root of `f`, then `f(g(x))` is irreducible. -/
+theorem _root_.Polynomial.irreducible_comp {f g : K[X]} (hfm : f.Monic) (hgm : g.Monic)
+ (hf : Irreducible f)
+ (hg : ∀ (E : Type u) [Field E] [Algebra K E] (x : E) (hx : minpoly K x = f),
+ Irreducible (g.map (algebraMap _ _) - C (AdjoinSimple.gen K x))) :
+ Irreducible (f.comp g) := by
+ have hf' : natDegree f ≠ 0 :=
+ fun e ↦ not_irreducible_C (f.coeff 0) (eq_C_of_natDegree_eq_zero e ▸ hf)
+ have hg' : natDegree g ≠ 0
+ · have := Fact.mk hf
+ intro e
+ apply not_irreducible_C ((g.map (algebraMap _ _)).coeff 0 - AdjoinSimple.gen K (root f))
+ rw [map_sub, coeff_map, ← map_C, ← eq_C_of_natDegree_eq_zero e]
+ apply hg (AdjoinRoot f)
+ rw [AdjoinRoot.minpoly_root hf.ne_zero, hfm, inv_one, map_one, mul_one]
+ have H₁ : f.comp g ≠ 0 := fun h ↦ by simpa [hf', hg', natDegree_comp] using congr_arg natDegree h
+ have H₂ : ¬ IsUnit (f.comp g) := fun h ↦
+ by simpa [hf', hg', natDegree_comp] using natDegree_eq_zero_of_isUnit h
+ have ⟨p, hp₁, hp₂⟩ := WfDvdMonoid.exists_irreducible_factor H₂ H₁
+ suffices natDegree p = natDegree f * natDegree g from (associated_of_dvd_of_natDegree_le hp₂ H₁
+ (this.trans natDegree_comp.symm).ge).irreducible hp₁
+ have := Fact.mk hp₁
+ let Kx := AdjoinRoot p
+ letI := (AdjoinRoot.powerBasis hp₁.ne_zero).finiteDimensional
+ have key₁ : f = minpoly K (aeval (root p) g)
+ · refine minpoly.eq_of_irreducible_of_monic hf ?_ hfm
+ rw [← aeval_comp]
+ exact aeval_eq_zero_of_dvd_aeval_eq_zero hp₂ (AdjoinRoot.eval₂_root p)
+ have key₁' : finrank K K⟮aeval (root p) g⟯ = natDegree f
+ · rw [adjoin.finrank, ← key₁]
+ exact IsIntegral.of_finite _ _
+ have key₂ : g.map (algebraMap _ _) - C (AdjoinSimple.gen K (aeval (root p) g)) =
+ minpoly K⟮aeval (root p) g⟯ (root p)
+ · exact minpoly.eq_of_irreducible_of_monic (hg _ _ key₁.symm) (by simp [AdjoinSimple.gen])
+ (Monic.sub_of_left (hgm.map _) (degree_lt_degree (by simpa [Nat.pos_iff_ne_zero] using hg')))
+ have key₂' : finrank K⟮aeval (root p) g⟯ Kx = natDegree g
+ · trans natDegree (minpoly K⟮aeval (root p) g⟯ (root p))
+ · have : K⟮aeval (root p) g⟯⟮root p⟯ = ⊤
+ · apply restrictScalars_injective K
+ rw [restrictScalars_top, adjoin_adjoin_left, Set.union_comm, ← adjoin_adjoin_left,
+ adjoin_root_eq_top p, restrictScalars_adjoin]
+ simp
+ rw [← finrank_top', ← this, adjoin.finrank]
+ exact IsIntegral.of_finite _ _
+ · simp [← key₂]
+ have := FiniteDimensional.finrank_mul_finrank K K⟮aeval (root p) g⟯ Kx
+ rwa [key₁', key₂', (AdjoinRoot.powerBasis hp₁.ne_zero).finrank, powerBasis_dim, eq_comm] at this
+
end AdjoinIntegralElement
section Induction
feat(FieldTheory/SeparableClosure): (relative) separable closure (#9338)
Main definitions:
separableClosure: the (relative) separable closure ofE / F, or called maximal separable subextension ofE / F, is defined to be the intermediate field ofE / Fconsisting of all separable elements.Field.sepDegree F E: the (infinite) separable degree[E:F]_sof an algebraic extensionE / Fof fields, defined to be the degree ofseparableClosure F E / F.Field.insepDegree F E: the (infinite) inseparable degree[E:F]_iof an algebraic extensionE / Fof fields, defined to be the degree ofE / separableClosure F E.Field.finInsepDegree F E: the (finite) inseparable degree[E:F]_iof an algebraic extensionE / Fof fields, defined to be the degree ofE / separableClosure F Eas a natural number. It is zero if such field extension is not finite.
Main results:
le_separableClosure_iff: an intermediate field ofE / Fis contained in the (relative) separable closure ofE / Fif and only if it is separable overF.separableClosure.normalClosure_eq_self: the normal closure of the (relative) separable closure ofE / Fis equal to itself.separableClosure.normal: the (relative) separable closure of a normal extension is normal.separableClosure.isSepClosure: the (relative) separable closure of a separably closed extension is a separable closure of the base field.IntermediateField.isSeparable_adjoin_iff_separable:F(S) / Fis a separable extension if and only if all elements ofSare separable elements.separableClosure.eq_top_iff: the (relative) separable closure ofE / Fis equal toEif and only ifE / Fis separable.
Diff
@@ -102,6 +102,10 @@ instance : CompleteLattice (IntermediateField F E) where
instance : Inhabited (IntermediateField F E) :=
⟨⊤⟩
+instance : Unique (IntermediateField F F) :=
+ { inferInstanceAs (Inhabited (IntermediateField F F)) with
+ uniq := fun _ ↦ toSubalgebra_injective <| Subsingleton.elim _ _ }
+
theorem coe_bot : ↑(⊥ : IntermediateField F E) = Set.range (algebraMap F E) := rfl
#align intermediate_field.coe_bot IntermediateField.coe_bot
chore: Improve Finset lemma names (#8894)
Change a few lemma names that have historically bothered me.
Finset.card_le_of_subset→Finset.card_le_cardMultiset.card_le_of_le→Multiset.card_le_cardMultiset.card_lt_of_lt→Multiset.card_lt_cardSet.ncard_le_of_subset→Set.ncard_le_ncardFinset.image_filter→Finset.filter_imageCompleteLattice.finset_sup_compact_of_compact→CompleteLattice.isCompactElement_finset_sup
Diff
@@ -749,7 +749,7 @@ theorem adjoin_finset_isCompactElement (S : Finset E) :
IsCompactElement (adjoin F S : IntermediateField F E) := by
rw [← biSup_adjoin_simple]
simp_rw [Finset.mem_coe, ← Finset.sup_eq_iSup]
- exact finset_sup_compact_of_compact S fun x _ => adjoin_simple_isCompactElement x
+ exact isCompactElement_finsetSup S fun x _ => adjoin_simple_isCompactElement x
#align intermediate_field.adjoin_finset_is_compact_element IntermediateField.adjoin_finset_isCompactElement
/-- Adjoining a finite subset is compact in the lattice of intermediate fields. -/
feat(FieldTheory/Adjoin): add exists_lt_finrank_of_infinite_dimensional (#9262)
... which state that an infinite algebraic extension has finite subextension with arbitrary large degree.
Diff
@@ -1043,6 +1043,26 @@ theorem adjoin_minpoly_coeff_of_exists_primitive_element
((g.monic_toSubring _ _).mpr <| (minpoly.monic <| .of_finite K α).map _).ne_zero using 1
rw [natDegree_toSubring, natDegree_map]
+variable {F} in
+/-- If `E / F` is an infinite algebraic extension, then there exists an intermediate field
+`L / F` with arbitrarily large finite extension degree. -/
+theorem exists_lt_finrank_of_infinite_dimensional
+ (halg : Algebra.IsAlgebraic F E) (hnfd : ¬ FiniteDimensional F E) (n : ℕ) :
+ ∃ L : IntermediateField F E, FiniteDimensional F L ∧ n < finrank F L := by
+ induction' n with n ih
+ · exact ⟨⊥, Subalgebra.finiteDimensional_bot, finrank_pos⟩
+ obtain ⟨L, fin, hn⟩ := ih
+ obtain ⟨x, hx⟩ : ∃ x : E, x ∉ L := by
+ contrapose! hnfd
+ rw [show L = ⊤ from eq_top_iff.2 fun x _ ↦ hnfd x] at fin
+ exact topEquiv.toLinearEquiv.finiteDimensional
+ let L' := L ⊔ F⟮x⟯
+ haveI := adjoin.finiteDimensional (halg x).isIntegral
+ refine ⟨L', inferInstance, by_contra fun h ↦ ?_⟩
+ have h1 : L = L' := eq_of_le_of_finrank_le le_sup_left ((not_lt.1 h).trans hn)
+ have h2 : F⟮x⟯ ≤ L' := le_sup_right
+ exact hx <| (h1.symm ▸ h2) <| mem_adjoin_simple_self F x
+
theorem _root_.minpoly.natDegree_le (x : L) [FiniteDimensional K L] :
(minpoly K x).natDegree ≤ finrank K L :=
le_of_eq_of_le (IntermediateField.adjoin.finrank (.of_finite _ _)).symm
chore: cleanups following #8609 and #8714 (#8962)
-
generalize
image_rootSet,adjoin_rootSet_eq_rangeandsplits_comp_of_splitsin Data/Polynomial/Splits and use the last one to golfsplits_of_algHom,splits_of_isScalarTower(introduced in # 8609). -
add three new lemmas
mem_range_x_of_minpoly_splitsto simplify the construction ofIntermediateField.algHomEquivAlgHomOfIsAlgClosedandAlgebra.IsAlgebraic.algHomEquivAlgHomOfIsAlgClosed, remove theIsAlgClosedcondition and rename.They could be moved to an earlier file but I refrain from doing that. (#find_home says it's already in the right place) -
golf
primitive_element_iff_algHom_eq_of_evalfrom # 8609, using a new lemmaIsIntegral.minpoly_splits_tower_topfor the last step. -
make
integralClosure_algEquiv_restrict(from # 8714) computable and rename toAlgEquiv.mapIntegralClosureto follow camelCase naming convention and enable dot notation.
Co-authored-by: Xavier-François Roblot <46200072+xroblot@users.noreply.github.com> Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>
Diff
@@ -848,14 +848,20 @@ theorem finrank_eq_one_iff : finrank F K = 1 ↔ K = ⊥ := by
bot_toSubalgebra]
#align intermediate_field.finrank_eq_one_iff IntermediateField.finrank_eq_one_iff
-@[simp]
+@[simp] protected
theorem rank_bot : Module.rank F (⊥ : IntermediateField F E) = 1 := by rw [rank_eq_one_iff]
#align intermediate_field.rank_bot IntermediateField.rank_bot
-@[simp]
+@[simp] protected
theorem finrank_bot : finrank F (⊥ : IntermediateField F E) = 1 := by rw [finrank_eq_one_iff]
#align intermediate_field.finrank_bot IntermediateField.finrank_bot
+@[simp] protected theorem rank_top : Module.rank (⊤ : IntermediateField F E) E = 1 :=
+ Subalgebra.bot_eq_top_iff_rank_eq_one.mp <| top_le_iff.mp fun x _ ↦ ⟨⟨x, trivial⟩, rfl⟩
+
+@[simp] protected theorem finrank_top : finrank (⊤ : IntermediateField F E) E = 1 :=
+ rank_eq_one_iff_finrank_eq_one.mp IntermediateField.rank_top
+
theorem rank_adjoin_eq_one_iff : Module.rank F (adjoin F S) = 1 ↔ S ⊆ (⊥ : IntermediateField F E) :=
Iff.trans rank_eq_one_iff adjoin_eq_bot_iff
#align intermediate_field.rank_adjoin_eq_one_iff IntermediateField.rank_adjoin_eq_one_iff
feat: Add some equivalent characterisations of primitive elements in finite extensions of fields (#8609)
See the Zulip thread
Diff
@@ -1296,6 +1296,25 @@ end AlgHomMkAdjoinSplits
end IntermediateField
+section Algebra.IsAlgebraic
+
+/-- Let `K/F` be an algebraic extension of fields and `L` a field in which all the minimal
+polynomial over `F` of elements of `K` splits. Then, for `x ∈ K`, the images of `x` by the
+`F`-algebra morphisms from `K` to `L` are exactly the roots in `L` of the minimal polynomial
+of `x` over `F`. -/
+theorem Algebra.IsAlgebraic.range_eval_eq_rootSet_minpoly_of_splits {F K : Type*} (L : Type*)
+ [Field F] [Field K] [Field L] [Algebra F L] [Algebra F K]
+ (hA : ∀ x : K, (minpoly F x).Splits (algebraMap F L))
+ (hK : Algebra.IsAlgebraic F K) (x : K) :
+ (Set.range fun (ψ : K →ₐ[F] L) => ψ x) = (minpoly F x).rootSet L := by
+ ext a
+ rw [mem_rootSet_of_ne (minpoly.ne_zero (hK.isIntegral x))]
+ refine ⟨fun ⟨ψ, hψ⟩ ↦ ?_, fun ha ↦ IntermediateField.exists_algHom_of_splits_of_aeval
+ (fun x ↦ ⟨hK.isIntegral x, hA x⟩) ha⟩
+ rw [← hψ, Polynomial.aeval_algHom_apply ψ x, minpoly.aeval, map_zero]
+
+end Algebra.IsAlgebraic
+
section PowerBasis
variable {K L : Type*} [Field K] [Field L] [Algebra K L]
Diff
@@ -452,7 +452,7 @@ theorem lift_bot (K : IntermediateField F E) :
@[simp]
theorem lift_top (K : IntermediateField F E) :
- lift (F := K) ⊤ = K := by rw [lift, ←AlgHom.fieldRange_eq_map, fieldRange_val]
+ lift (F := K) ⊤ = K := by rw [lift, ← AlgHom.fieldRange_eq_map, fieldRange_val]
@[simp]
theorem adjoin_self (K : IntermediateField F E) :
feat: IsNormalClosure predicate (#8418)
Main changes are to the file NormalClosure:
-
Introduce predicate
IsNormalClosureto characterize normal closures L/F of a field extension K/F by the conditions that every minimal polynomial of an element of K over F splits in L, and that L is generated by roots of such polynomials. (When K/F is not necessarily algebraic, the conditions actually says L/F is a normal closure of the algebraic closure of F in K over F. -
IsNormalClosure.normal: a normal closure is always normal. -
isNormalClosure_iff: if K/F is algebraic, the "generated by roots" condition in IsNormalClosure can be replaced by "generated by images of embeddings". To prove it, we split out the two inclusions inrestrictScalars_eq_iSup_adjoinand golf its proof.restrictScalars_eq_iSup_adjoinis renamed tonormalClosure_eq_iSup_adjoinas it has nothing to do withrestrictScalars. -
IsNormalClosure.lift: a normal closure of K/F embeds into any L/F such that the minpolys of K/F splits in L/F. -
IsNormalClosure.equiv: normal closures are unique up to F-algebra isomorphisms. -
isNormalClosure_normalClosure:normalClosure F K Lis a valid normal closure if K/F is algebraic and all minpolys of K/F splits in L/F; in particular, if there is at least one F-embedding of K into L, and L/F is normal. -
Algebra.IsAlgebraic.cardinal_mk_algHom_le_of_splits: if every minpoly ofK/Fsplits inL/F, thenLis maximal w.r.t.F-embeddings ofK, in the sense thatK →ₐ[F] Lachieves maximal cardinality.
In the file Normal:
splits_of_mem_adjoin: If a set of algebraic elements in a field extensionK/Fhave minimal polynomials that split in another extensionL/F, then all minimal polynomials in the intermediate field generated by the set also split inL/F. This is in preparation for connectingIsNormalClosureandIsSplittingField.
In the file IntermediateField:
- Add
comapand show it forms a Galois connection withmap.
In the file FieldTheory/Adjoin:
- Add
map_sup/iSuplemmas that follow from the Galois connection, plus an additional convenience lemma.
In the file RingTheory/Algebraic: add a lemma AlgHom.isAlgebraic_of_injective.
Co-authored-by: Jz Pan <acme_pjz@hotmail.com>
Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>
Diff
@@ -209,7 +209,7 @@ theorem equivOfEq_rfl (S : IntermediateField F E) : equivOfEq (rfl : S = S) = Al
@[simp]
theorem equivOfEq_trans {S T U : IntermediateField F E} (hST : S = T) (hTU : T = U) :
- (equivOfEq hST).trans (equivOfEq hTU) = equivOfEq (_root_.trans hST hTU) :=
+ (equivOfEq hST).trans (equivOfEq hTU) = equivOfEq (hST.trans hTU) :=
rfl
#align intermediate_field.equiv_of_eq_trans IntermediateField.equivOfEq_trans
@@ -274,28 +274,37 @@ theorem restrictScalars_top {K : Type*} [Field K] [Algebra K E] [Algebra K F]
rfl
#align intermediate_field.restrict_scalars_top IntermediateField.restrictScalars_top
+variable {K : Type*} [Field K] [Algebra F K]
+
@[simp]
-theorem map_bot {K : Type*} [Field K] [Algebra F K] (f : E →ₐ[F] K) :
+theorem map_bot (f : E →ₐ[F] K) :
IntermediateField.map f ⊥ = ⊥ :=
toSubalgebra_injective <| Algebra.map_bot _
-theorem _root_.AlgHom.fieldRange_eq_map {K : Type*} [Field K] [Algebra F K] (f : E →ₐ[F] K) :
+theorem map_sup (s t : IntermediateField F E) (f : E →ₐ[F] K) : (s ⊔ t).map f = s.map f ⊔ t.map f :=
+ (gc_map_comap f).l_sup
+
+theorem map_iSup {ι : Sort*} (f : E →ₐ[F] K) (s : ι → IntermediateField F E) :
+ (iSup s).map f = ⨆ i, (s i).map f :=
+ (gc_map_comap f).l_iSup
+
+theorem _root_.AlgHom.fieldRange_eq_map (f : E →ₐ[F] K) :
f.fieldRange = IntermediateField.map f ⊤ :=
SetLike.ext' Set.image_univ.symm
#align alg_hom.field_range_eq_map AlgHom.fieldRange_eq_map
-theorem _root_.AlgHom.map_fieldRange {K L : Type*} [Field K] [Field L] [Algebra F K] [Algebra F L]
+theorem _root_.AlgHom.map_fieldRange {L : Type*} [Field L] [Algebra F L]
(f : E →ₐ[F] K) (g : K →ₐ[F] L) : f.fieldRange.map g = (g.comp f).fieldRange :=
SetLike.ext' (Set.range_comp g f).symm
#align alg_hom.map_field_range AlgHom.map_fieldRange
-theorem _root_.AlgHom.fieldRange_eq_top {K : Type*} [Field K] [Algebra F K] {f : E →ₐ[F] K} :
+theorem _root_.AlgHom.fieldRange_eq_top {f : E →ₐ[F] K} :
f.fieldRange = ⊤ ↔ Function.Surjective f :=
SetLike.ext'_iff.trans Set.range_iff_surjective
#align alg_hom.field_range_eq_top AlgHom.fieldRange_eq_top
@[simp]
-theorem _root_.AlgEquiv.fieldRange_eq_top {K : Type*} [Field K] [Algebra F K] (f : E ≃ₐ[F] K) :
+theorem _root_.AlgEquiv.fieldRange_eq_top (f : E ≃ₐ[F] K) :
(f : E →ₐ[F] K).fieldRange = ⊤ :=
AlgHom.fieldRange_eq_top.mpr f.surjective
#align alg_equiv.field_range_eq_top AlgEquiv.fieldRange_eq_top
@@ -371,8 +380,8 @@ theorem adjoin_le_subfield {K : Subfield E} (HF : Set.range (algebraMap F E) ⊆
theorem adjoin_subset_adjoin_iff {F' : Type*} [Field F'] [Algebra F' E] {S S' : Set E} :
(adjoin F S : Set E) ⊆ adjoin F' S' ↔
Set.range (algebraMap F E) ⊆ adjoin F' S' ∧ S ⊆ adjoin F' S' :=
- ⟨fun h => ⟨_root_.trans (adjoin.range_algebraMap_subset _ _) h, _root_.trans
- (subset_adjoin _ _) h⟩, fun ⟨hF, hS⟩ =>
+ ⟨fun h => ⟨(adjoin.range_algebraMap_subset _ _).trans h,
+ (subset_adjoin _ _).trans h⟩, fun ⟨hF, hS⟩ =>
(Subfield.closure_le (t := (adjoin F' S').toSubfield)).mpr (Set.union_subset hF hS)⟩
#align intermediate_field.adjoin_subset_adjoin_iff IntermediateField.adjoin_subset_adjoin_iff
@@ -581,19 +590,27 @@ theorem adjoin_simple_toSubalgebra_of_integral (hα : IsIntegral F α) :
rwa [isAlgebraic_iff_isIntegral]
#align intermediate_field.adjoin_simple_to_subalgebra_of_integral IntermediateField.adjoin_simple_toSubalgebra_of_integral
+/-- Characterize `IsSplittingField` with `IntermediateField.adjoin` instead of `Algebra.adjoin`. -/
+theorem _root_.isSplittingField_iff_intermediateField {p : F[X]} :
+ p.IsSplittingField F E ↔ p.Splits (algebraMap F E) ∧ adjoin F (p.rootSet E) = ⊤ := by
+ rw [← toSubalgebra_injective.eq_iff,
+ adjoin_algebraic_toSubalgebra fun _ ↦ isAlgebraic_of_mem_rootSet]
+ exact ⟨fun ⟨spl, adj⟩ ↦ ⟨spl, adj⟩, fun ⟨spl, adj⟩ ↦ ⟨spl, adj⟩⟩
+
+-- Note: p.Splits (algebraMap F E) also works
theorem isSplittingField_iff {p : F[X]} {K : IntermediateField F E} :
p.IsSplittingField F K ↔ p.Splits (algebraMap F K) ∧ K = adjoin F (p.rootSet E) := by
suffices _ → (Algebra.adjoin F (p.rootSet K) = ⊤ ↔ K = adjoin F (p.rootSet E)) by
- exact ⟨fun h => ⟨h.1, (this h.1).mp h.2⟩, fun h => ⟨h.1, (this h.1).mpr h.2⟩⟩
- simp_rw [SetLike.ext_iff, ← mem_toSubalgebra, ← SetLike.ext_iff]
- rw [adjoin_algebraic_toSubalgebra fun x => isAlgebraic_of_mem_rootSet]
- refine' fun hp => (adjoin_rootSet_eq_range hp K.val).symm.trans _
+ exact ⟨fun h ↦ ⟨h.1, (this h.1).mp h.2⟩, fun h ↦ ⟨h.1, (this h.1).mpr h.2⟩⟩
+ rw [← toSubalgebra_injective.eq_iff,
+ adjoin_algebraic_toSubalgebra fun x ↦ isAlgebraic_of_mem_rootSet]
+ refine fun hp ↦ (adjoin_rootSet_eq_range hp K.val).symm.trans ?_
rw [← K.range_val, eq_comm]
#align intermediate_field.is_splitting_field_iff IntermediateField.isSplittingField_iff
theorem adjoin_rootSet_isSplittingField {p : F[X]} (hp : p.Splits (algebraMap F E)) :
p.IsSplittingField F (adjoin F (p.rootSet E)) :=
- isSplittingField_iff.mpr ⟨splits_of_splits hp fun _ hx => subset_adjoin F (p.rootSet E) hx, rfl⟩
+ isSplittingField_iff.mpr ⟨splits_of_splits hp fun _ hx ↦ subset_adjoin F (p.rootSet E) hx, rfl⟩
#align intermediate_field.adjoin_root_set_is_splitting_field IntermediateField.adjoin_rootSet_isSplittingField
open scoped BigOperators
@@ -714,9 +731,8 @@ has been improved. -/
theorem adjoin_simple_le_iff {K : IntermediateField F E} : F⟮α⟯ ≤ K ↔ α ∈ K := by simp
#align intermediate_field.adjoin_simple_le_iff IntermediateField.adjoin_simple_le_iff
-theorem biSup_adjoin_simple : ⨆ x ∈ S, F⟮x⟯ = adjoin F S :=
- le_antisymm (iSup_le fun _ ↦ iSup_le fun hx ↦ adjoin_simple_le_iff.mpr <| subset_adjoin F S hx) <|
- adjoin_le_iff.mpr fun x hx ↦ adjoin_simple_le_iff.mp (le_iSup_of_le x (le_iSup_of_le hx le_rfl))
+theorem biSup_adjoin_simple : ⨆ x ∈ S, F⟮x⟯ = adjoin F S := by
+ rw [← iSup_subtype'', ← gc.l_iSup, iSup_subtype'']; congr; exact S.biUnion_of_singleton
/-- Adjoining a single element is compact in the lattice of intermediate fields. -/
theorem adjoin_simple_isCompactElement (x : E) : IsCompactElement F⟮x⟯ := by
@@ -751,8 +767,7 @@ instance : IsCompactlyGenerated (IntermediateField F E) :=
theorem exists_finset_of_mem_iSup {ι : Type*} {f : ι → IntermediateField F E} {x : E}
(hx : x ∈ ⨆ i, f i) : ∃ s : Finset ι, x ∈ ⨆ i ∈ s, f i := by
- have := (adjoin_simple_isCompactElement x : IsCompactElement F⟮x⟯).exists_finset_of_le_iSup
- (IntermediateField F E) f
+ have := (adjoin_simple_isCompactElement x).exists_finset_of_le_iSup (IntermediateField F E) f
simp only [adjoin_simple_le_iff] at this
exact this hx
#align intermediate_field.exists_finset_of_mem_supr IntermediateField.exists_finset_of_mem_iSup
chore(IntegralClosure): noncommutative generalizations and golfs (#8406)
Initially I just wanted to add more dot notations for IsIntegral and IsAlgebraic (done in #8437); then I noticed near-duplicates
Algebra.isIntegral_of_finite [Field R] [Ring A] and
RingHom.IsIntegral.of_finite [CommRing R] [CommRing A]
so I went on to generalize the latter to cover the former, and generalized everything in the IntegralClosure file to the noncommutative case whenever possible.
In the process I noticed more golfs, which result in this PR. Most notably, isIntegral_of_mem_of_FG is now proven using Cayley-Hamilton and doesn't depend on the Noetherian case isIntegral_of_noetherian; the latter is now proven using the former. In total the golfs makes mathlib 227 lines leaner (+487 -714).
The main changes are in the single file RingTheory/IntegralClosure:
-
Change the definition of
Algebra.IsIntegralwhich makes it unfold toIsIntegralrather thanRingHom.IsIntegralElembecause the former has much more APIs. -
Fix lemma names involving
is_integralwhich are actually aboutIsIntegralElem:RingHom.is_integral_map→RingHom.isIntegralElem_mapRingHom.is_integral_of_mem_closure→RingHom.IsIntegralElem.of_mem_closureRingHom.is_integral_zero/one→RingHom.isIntegralElem_zero/oneRingHom.is_integral_add/neg/sub/mul/of_mul_unit→RingHom.IsIntegralElem.add/neg/sub/mul/of_mul_unit -
Add a lemma
Algebra.IsIntegral.of_injective. -
Move
isIntegral_of_(submodule_)noetheriandown and golf them. -
Remove
(Algebra.)isIntegral_of_finitethat work only over fields, in favor of the more general(Algebra.)isIntegral.of_finite. -
Merge duplicate lemmas
isIntegral_of_isScalarTowerandisIntegral_tower_top_of_isIntegralintoIsIntegral.tower_top. -
Golf
IsIntegral.of_mem_of_fgby first provingIsIntegral.of_finiteusing Cayley-Hamilton. -
Add a docstring mentioning the Kurosh problem at
Algebra.IsIntegral.finite. The negative solution to the problem means the theorem doesn't generalize to noncommutative algebras. -
Golf
IsIntegral.tmulandisField_of_isIntegral_of_isField('). -
Combine
isIntegral_trans_auxintoisIntegral_transand golf. -
Add
Algebranamespace toisIntegral_sup. -
rename lemmas for dot notation:
RingHom.isIntegral_trans→RingHom.IsIntegral.transRingHom.isIntegral_quotient/tower_bot/top_of_isIntegral→RingHom.IsIntegral.quotient/tower_bot/topisIntegral_of_mem_closure'→IsIntegral.of_mem_closure'(and the '' version)isIntegral_of_surjective→Algebra.isIntegral_of_surjective
The next changed file is RingTheory/Algebraic:
-
Rename:
of_larger_base→tower_top(for consistency withIsIntegral)Algebra.isAlgebraic_of_finite→Algebra.IsAlgebraic.of_finiteAlgebra.isAlgebraic_trans→Algebra.IsAlgebraic.trans -
Add new lemmas
Algebra.IsIntegral.isAlgebraic,isAlgebraic_algHom_iff, andAlgebra.IsAlgebraic.of_injectiveto streamline some proofs.
The generalization from CommRing to Ring requires an additional lemma scaleRoots_eval₂_mul_of_commute in Polynomial/ScaleRoots.
A lemma Algebra.lmul_injective is added to Algebra/Bilinear (in order to golf the proof of IsIntegral.of_mem_of_fg).
In all other files, I merely fix the changed names, or use newly available dot notations.
Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>
Diff
@@ -624,7 +624,8 @@ theorem sup_toSubalgebra [h1 : FiniteDimensional K E1] [h2 : FiniteDimensional K
exact (congr_arg (· ∈ S1 ⊔ S2) <| eq_inv_of_mul_eq_one_right <| Subtype.ext_iff.mp h).mp y.2
exact
isField_of_isIntegral_of_isField'
- (isIntegral_sup.mpr ⟨Algebra.isIntegral_of_finite K E1, Algebra.isIntegral_of_finite K E2⟩)
+ (Algebra.isIntegral_sup.mpr
+ ⟨Algebra.IsIntegral.of_finite K E1, Algebra.IsIntegral.of_finite K E2⟩)
(Field.toIsField K)
#align intermediate_field.sup_to_subalgebra IntermediateField.sup_toSubalgebra
@@ -773,7 +774,7 @@ theorem exists_finset_of_mem_supr'' {ι : Type*} {f : ι → IntermediateField F
(subset_adjoin F (rootSet (minpoly F x1) E) _)
· rw [IntermediateField.minpoly_eq, Subtype.coe_mk]
· rw [mem_rootSet_of_ne, minpoly.aeval]
- exact minpoly.ne_zero (isIntegral_iff.mp (isAlgebraic_iff_isIntegral.mp (h i ⟨x1, hx1⟩)))
+ exact minpoly.ne_zero (isIntegral_iff.mp (h i ⟨x1, hx1⟩).isIntegral)
#align intermediate_field.exists_finset_of_mem_supr'' IntermediateField.exists_finset_of_mem_supr''
end AdjoinSimple
@@ -977,7 +978,7 @@ theorem adjoin.finiteDimensional {x : L} (hx : IsIntegral K x) : FiniteDimension
#align intermediate_field.adjoin.finite_dimensional IntermediateField.adjoin.finiteDimensional
theorem isAlgebraic_adjoin_simple {x : L} (hx : IsIntegral K x) : Algebra.IsAlgebraic K K⟮x⟯ :=
- have := adjoin.finiteDimensional hx; Algebra.isAlgebraic_of_finite K K⟮x⟯
+ have := adjoin.finiteDimensional hx; Algebra.IsAlgebraic.of_finite K K⟮x⟯
theorem adjoin.finrank {x : L} (hx : IsIntegral K x) :
FiniteDimensional.finrank K K⟮x⟯ = (minpoly K x).natDegree := by
@@ -1006,31 +1007,24 @@ theorem adjoin_minpoly_coeff_of_exists_primitive_element
rintro ⟨n, -, rfl⟩
rw [coeff_map]
apply Subtype.mem
- have g_lifts : g ∈ lifts (algebraMap K' E) := by
- refine g.lifts_iff_coeff_lifts.mpr fun n ↦ ?_
- erw [Subtype.range_val]
- by_cases hn : n ∈ g.support
- · exact subset_adjoin F _ (mem_frange_iff.mpr ⟨n, hn, rfl⟩)
- · exact not_mem_support_iff.mp hn ▸ zero_mem K'
- obtain ⟨p, hp⟩ := g.lifts_and_natDegree_eq_and_monic
- g_lifts ((minpoly.monic <| IsIntegral.of_finite K α).map _)
- have dvd_p : minpoly K' α ∣ p
+ have dvd_g : minpoly K' α ∣ g.toSubring K'.toSubring (subset_adjoin F _)
· apply minpoly.dvd
- rw [aeval_def, eval₂_eq_eval_map, hp.1, ← eval₂_eq_eval_map, ← aeval_def]
+ erw [aeval_def, eval₂_eq_eval_map, g.map_toSubring K'.toSubring, eval_map, ← aeval_def]
exact minpoly.aeval K α
have finrank_eq : ∀ K : IntermediateField F E, finrank K E = natDegree (minpoly K α)
· intro K
- have := adjoin.finrank (IsIntegral.of_finite K α)
+ have := adjoin.finrank (.of_finite K α)
erw [adjoin_eq_top_of_adjoin_eq_top F hprim, finrank_top K E] at this
exact this
refine eq_of_le_of_finrank_le' hsub ?_
simp_rw [finrank_eq]
- convert natDegree_le_of_dvd dvd_p hp.2.2.ne_zero using 1
- rw [hp.2.1, natDegree_map]
+ convert natDegree_le_of_dvd dvd_g
+ ((g.monic_toSubring _ _).mpr <| (minpoly.monic <| .of_finite K α).map _).ne_zero using 1
+ rw [natDegree_toSubring, natDegree_map]
theorem _root_.minpoly.natDegree_le (x : L) [FiniteDimensional K L] :
(minpoly K x).natDegree ≤ finrank K L :=
- le_of_eq_of_le (IntermediateField.adjoin.finrank (IsIntegral.of_finite _ _)).symm
+ le_of_eq_of_le (IntermediateField.adjoin.finrank (.of_finite _ _)).symm
K⟮x⟯.toSubmodule.finrank_le
#align minpoly.nat_degree_le minpoly.natDegree_le
@@ -1047,8 +1041,8 @@ theorem isAlgebraic_iSup {ι : Type*} {t : ι → IntermediateField K L}
obtain ⟨s, hx⟩ := exists_finset_of_mem_supr' hx
rw [isAlgebraic_iff, Subtype.coe_mk, ← Subtype.coe_mk (p := (· ∈ _)) x hx, ← isAlgebraic_iff]
haveI : ∀ i : Σ i, t i, FiniteDimensional K K⟮(i.2 : L)⟯ := fun ⟨i, x⟩ ↦
- adjoin.finiteDimensional (isIntegral_iff.1 (isAlgebraic_iff_isIntegral.1 (h i x)))
- apply Algebra.isAlgebraic_of_finite
+ adjoin.finiteDimensional (isIntegral_iff.1 (h i x).isIntegral)
+ apply Algebra.IsAlgebraic.of_finite
#align intermediate_field.is_algebraic_supr IntermediateField.isAlgebraic_iSup
theorem isAlgebraic_adjoin {S : Set L} (hS : ∀ x ∈ S, IsIntegral K x) :
@@ -1211,7 +1205,7 @@ theorem Lifts.exists_upper_bound (c : Set (Lifts F E K)) (hc : IsChain (· ≤
whose carrier contains `s`. -/
theorem Lifts.exists_lift_of_splits (x : Lifts F E K) {s : E} (h1 : IsIntegral F s)
(h2 : (minpoly F s).Splits (algebraMap F K)) : ∃ y, x ≤ y ∧ s ∈ y.carrier :=
- have I1 : IsIntegral x.carrier s := isIntegral_of_isScalarTower h1
+ have I1 : IsIntegral x.carrier s := h1.tower_top
have I2 := (minpoly.degree_pos I1).ne'
have key : (minpoly x.carrier s).Splits x.emb.toRingHom :=
splits_of_splits_of_dvd _ (map_ne_zero (minpoly.ne_zero h1))
chore(RingTheory/{Algebraic, Localization/Integral}): rename decls to use dot notation (#8437)
This PR tests a string-based tool for renaming declarations.
Inspired by this Zulip thread, I am trying to reduce the diff of #8406.
This PR makes the following renames:
| From | To |
Diff
@@ -1013,14 +1013,14 @@ theorem adjoin_minpoly_coeff_of_exists_primitive_element
· exact subset_adjoin F _ (mem_frange_iff.mpr ⟨n, hn, rfl⟩)
· exact not_mem_support_iff.mp hn ▸ zero_mem K'
obtain ⟨p, hp⟩ := g.lifts_and_natDegree_eq_and_monic
- g_lifts ((minpoly.monic <| isIntegral_of_finite K α).map _)
+ g_lifts ((minpoly.monic <| IsIntegral.of_finite K α).map _)
have dvd_p : minpoly K' α ∣ p
· apply minpoly.dvd
rw [aeval_def, eval₂_eq_eval_map, hp.1, ← eval₂_eq_eval_map, ← aeval_def]
exact minpoly.aeval K α
have finrank_eq : ∀ K : IntermediateField F E, finrank K E = natDegree (minpoly K α)
· intro K
- have := adjoin.finrank (isIntegral_of_finite K α)
+ have := adjoin.finrank (IsIntegral.of_finite K α)
erw [adjoin_eq_top_of_adjoin_eq_top F hprim, finrank_top K E] at this
exact this
refine eq_of_le_of_finrank_le' hsub ?_
@@ -1030,7 +1030,7 @@ theorem adjoin_minpoly_coeff_of_exists_primitive_element
theorem _root_.minpoly.natDegree_le (x : L) [FiniteDimensional K L] :
(minpoly K x).natDegree ≤ finrank K L :=
- le_of_eq_of_le (IntermediateField.adjoin.finrank (isIntegral_of_finite _ _)).symm
+ le_of_eq_of_le (IntermediateField.adjoin.finrank (IsIntegral.of_finite _ _)).symm
K⟮x⟯.toSubmodule.finrank_le
#align minpoly.nat_degree_le minpoly.natDegree_le
refactor: golf IntermediateField.Lifts (#8221)
Inspired by the IsAlgClosed.lift.SubfieldWithHom counterpart:
-
Change
Liftsfrom a Sigma type to a structure with fieldscarrierandemb. -
Change the definition of the partial order on
Liftsto useIntermediateField.inclusion. -
Use Subalgebra.iSupLift in the proof of
Lifts.exists_upper_bound.
Also:
-
Inline multiple auxiliary definitions for
Lifts.exists_upper_boundandLifts.exists_lift_of_splitsinto the proofs proper. -
Move the
Supremumsection much further up, in order to use the new lemmatoSubalgebra_iSup_of_directedto prove stuff aboutLifts(and golf a proof aboutCompactElement).isAlgebraic_iSuphowever can't be moved up, so I put it nearadjoin.finiteDimensional, the last lemma it depends on.
Co-authored-by: acmepjz <acme_pjz@hotmail.com>
Diff
@@ -96,10 +96,7 @@ instance : CompleteLattice (IntermediateField F E) where
__ := GaloisInsertion.liftCompleteLattice IntermediateField.gi
bot :=
{ toSubalgebra := ⊥
- inv_mem' := fun x (hx : x ∈ (⊥ : Subalgebra F E)) => show x⁻¹ ∈ (⊥ : Subalgebra F E) by
- rw [Algebra.mem_bot] at hx ⊢
- obtain ⟨r, rfl⟩ := hx
- exact ⟨r⁻¹, map_inv₀ _ _⟩ }
+ inv_mem' := by rintro x ⟨r, rfl⟩; exact ⟨r⁻¹, map_inv₀ _ _⟩ }
bot_le x := (bot_le : ⊥ ≤ x.toSubalgebra)
instance : Inhabited (IntermediateField F E) :=
@@ -193,19 +190,10 @@ theorem iInf_toSubfield {ι : Sort*} (S : ι → IntermediateField F E) :
SetLike.coe_injective <| by simp [iInf]
#align intermediate_field.infi_to_subfield IntermediateField.iInf_toSubfield
---Porting note: `left_inv`, `right_inv`, `map_mul'`, `map_add'` and `commutes` were not needed.
/-- Construct an algebra isomorphism from an equality of intermediate fields -/
-@[simps apply]
-def equivOfEq {S T : IntermediateField F E} (h : S = T) : S ≃ₐ[F] T := by
- refine'
- { toFun := fun x => ⟨x, _⟩
- invFun := fun x => ⟨x, _⟩
- left_inv := fun _ => by ext; simp
- right_inv := fun _ => by ext; simp
- map_mul' := fun _ _ => by ext; simp
- map_add' := fun _ _ => by ext; simp
- commutes' := fun _ => by ext; rfl } <;> aesop
-
+@[simps! apply]
+def equivOfEq {S T : IntermediateField F E} (h : S = T) : S ≃ₐ[F] T :=
+ Subalgebra.equivOfEq _ _ (congr_arg toSubalgebra h)
#align intermediate_field.equiv_of_eq IntermediateField.equivOfEq
@[simp]
@@ -488,6 +476,8 @@ theorem eq_adjoin_of_eq_algebra_adjoin (K : IntermediateField F E)
convert K.inv_mem (x := x) <;> rw [← h] <;> rfl
#align intermediate_field.eq_adjoin_of_eq_algebra_adjoin IntermediateField.eq_adjoin_of_eq_algebra_adjoin
+theorem adjoin_eq_top_of_algebra (hS : Algebra.adjoin F S = ⊤) : adjoin F S = ⊤ :=
+ top_le_iff.mp (hS.symm.trans_le <| algebra_adjoin_le_adjoin F S)
@[elab_as_elim]
theorem adjoin_induction {s : Set E} {p : E → Prop} {x} (h : x ∈ adjoin F s) (Hs : ∀ x ∈ s, p x)
@@ -608,22 +598,114 @@ theorem adjoin_rootSet_isSplittingField {p : F[X]} (hp : p.Splits (algebraMap F
open scoped BigOperators
+section Supremum
+
+variable {K L : Type*} [Field K] [Field L] [Algebra K L] (E1 E2 : IntermediateField K L)
+
+theorem le_sup_toSubalgebra : E1.toSubalgebra ⊔ E2.toSubalgebra ≤ (E1 ⊔ E2).toSubalgebra :=
+ sup_le (show E1 ≤ E1 ⊔ E2 from le_sup_left) (show E2 ≤ E1 ⊔ E2 from le_sup_right)
+#align intermediate_field.le_sup_to_subalgebra IntermediateField.le_sup_toSubalgebra
+
+theorem sup_toSubalgebra [h1 : FiniteDimensional K E1] [h2 : FiniteDimensional K E2] :
+ (E1 ⊔ E2).toSubalgebra = E1.toSubalgebra ⊔ E2.toSubalgebra := by
+ let S1 := E1.toSubalgebra
+ let S2 := E2.toSubalgebra
+ refine'
+ le_antisymm
+ (show _ ≤ (S1 ⊔ S2).toIntermediateField _ from
+ sup_le (show S1 ≤ _ from le_sup_left) (show S2 ≤ _ from le_sup_right))
+ (le_sup_toSubalgebra E1 E2)
+ suffices IsField (S1 ⊔ S2 : Subalgebra K L) by
+ intro x hx
+ by_cases hx' : (⟨x, hx⟩ : (S1 ⊔ S2 : Subalgebra K L)) = 0
+ · rw [← Subtype.coe_mk x, hx', Subalgebra.coe_zero, inv_zero]
+ exact (S1 ⊔ S2).zero_mem
+ · obtain ⟨y, h⟩ := this.mul_inv_cancel hx'
+ exact (congr_arg (· ∈ S1 ⊔ S2) <| eq_inv_of_mul_eq_one_right <| Subtype.ext_iff.mp h).mp y.2
+ exact
+ isField_of_isIntegral_of_isField'
+ (isIntegral_sup.mpr ⟨Algebra.isIntegral_of_finite K E1, Algebra.isIntegral_of_finite K E2⟩)
+ (Field.toIsField K)
+#align intermediate_field.sup_to_subalgebra IntermediateField.sup_toSubalgebra
+
+instance finiteDimensional_sup [h1 : FiniteDimensional K E1] [h2 : FiniteDimensional K E2] :
+ FiniteDimensional K (E1 ⊔ E2 : IntermediateField K L) := by
+ let g := Algebra.TensorProduct.productMap E1.val E2.val
+ suffices g.range = (E1 ⊔ E2).toSubalgebra by
+ have h : FiniteDimensional K (Subalgebra.toSubmodule g.range) :=
+ g.toLinearMap.finiteDimensional_range
+ rwa [this] at h
+ rw [Algebra.TensorProduct.productMap_range, E1.range_val, E2.range_val, sup_toSubalgebra]
+#align intermediate_field.finite_dimensional_sup IntermediateField.finiteDimensional_sup
+
+variable {ι : Type*} {t : ι → IntermediateField K L}
+
+theorem coe_iSup_of_directed [Nonempty ι] (dir : Directed (· ≤ ·) t) :
+ ↑(iSup t) = ⋃ i, (t i : Set L) :=
+ let M : IntermediateField K L :=
+ { __ := Subalgebra.copy _ _ (Subalgebra.coe_iSup_of_directed dir).symm
+ inv_mem' := fun _ hx ↦ have ⟨i, hi⟩ := Set.mem_iUnion.mp hx
+ Set.mem_iUnion.mpr ⟨i, (t i).inv_mem hi⟩ }
+ have : iSup t = M := le_antisymm
+ (iSup_le fun i ↦ le_iSup (fun i ↦ (t i : Set L)) i) (Set.iUnion_subset fun _ ↦ le_iSup t _)
+ this.symm ▸ rfl
+
+theorem toSubalgebra_iSup_of_directed (dir : Directed (· ≤ ·) t) :
+ (iSup t).toSubalgebra = ⨆ i, (t i).toSubalgebra := by
+ cases isEmpty_or_nonempty ι
+ · simp_rw [iSup_of_empty, bot_toSubalgebra]
+ · exact SetLike.ext' ((coe_iSup_of_directed dir).trans (Subalgebra.coe_iSup_of_directed dir).symm)
+
+instance finiteDimensional_iSup_of_finite [h : Finite ι] [∀ i, FiniteDimensional K (t i)] :
+ FiniteDimensional K (⨆ i, t i : IntermediateField K L) := by
+ rw [← iSup_univ]
+ let P : Set ι → Prop := fun s => FiniteDimensional K (⨆ i ∈ s, t i : IntermediateField K L)
+ change P Set.univ
+ apply Set.Finite.induction_on
+ all_goals dsimp only
+ · exact Set.finite_univ
+ · rw [iSup_emptyset]
+ exact (botEquiv K L).symm.toLinearEquiv.finiteDimensional
+ · intro _ s _ _ hs
+ rw [iSup_insert]
+ exact IntermediateField.finiteDimensional_sup _ _
+#align intermediate_field.finite_dimensional_supr_of_finite IntermediateField.finiteDimensional_iSup_of_finite
+
+instance finiteDimensional_iSup_of_finset
+ /-Porting note: changed `h` from `∀ i ∈ s, FiniteDimensional K (t i)` because this caused an
+ error. See `finiteDimensional_iSup_of_finset'` for a stronger version, that was the one
+ used in mathlib3.-/
+ {s : Finset ι} [∀ i, FiniteDimensional K (t i)] :
+ FiniteDimensional K (⨆ i ∈ s, t i : IntermediateField K L) :=
+ iSup_subtype'' s t ▸ IntermediateField.finiteDimensional_iSup_of_finite
+#align intermediate_field.finite_dimensional_supr_of_finset IntermediateField.finiteDimensional_iSup_of_finset
+
+theorem finiteDimensional_iSup_of_finset'
+ /-Porting note: this was the mathlib3 version. Using `[h : ...]`, as in mathlib3, causes the
+ error "invalid parametric local instance".-/
+ {s : Finset ι} (h : ∀ i, i ∈ s → FiniteDimensional K (t i)) :
+ FiniteDimensional K (⨆ i ∈ s, t i : IntermediateField K L) :=
+ have := Subtype.forall'.mp h
+ iSup_subtype'' s t ▸ IntermediateField.finiteDimensional_iSup_of_finite
+
/-- A compositum of splitting fields is a splitting field -/
-theorem isSplittingField_iSup {ι : Type*} {t : ι → IntermediateField F E} {p : ι → F[X]}
- {s : Finset ι} (h0 : ∏ i in s, p i ≠ 0) (h : ∀ i ∈ s, (p i).IsSplittingField F (t i)) :
- (∏ i in s, p i).IsSplittingField F (⨆ i ∈ s, t i : IntermediateField F E) := by
- let K : IntermediateField F E := ⨆ i ∈ s, t i
- have hK : ∀ i ∈ s, t i ≤ K := fun i hi => le_iSup_of_le i (le_iSup (fun _ => t i) hi)
+theorem isSplittingField_iSup {p : ι → K[X]}
+ {s : Finset ι} (h0 : ∏ i in s, p i ≠ 0) (h : ∀ i ∈ s, (p i).IsSplittingField K (t i)) :
+ (∏ i in s, p i).IsSplittingField K (⨆ i ∈ s, t i : IntermediateField K L) := by
+ let F : IntermediateField K L := ⨆ i ∈ s, t i
+ have hF : ∀ i ∈ s, t i ≤ F := fun i hi ↦ le_iSup_of_le i (le_iSup (fun _ ↦ t i) hi)
simp only [isSplittingField_iff] at h ⊢
refine'
- ⟨splits_prod (algebraMap F K) fun i hi =>
- Polynomial.splits_comp_of_splits (algebraMap F (t i)) (inclusion (hK i hi)).toRingHom
+ ⟨splits_prod (algebraMap K F) fun i hi ↦
+ splits_comp_of_splits (algebraMap K (t i)) (inclusion (hF i hi)).toRingHom
(h i hi).1,
_⟩
- simp only [rootSet_prod p s h0, ← Set.iSup_eq_iUnion, (@gc F _ E _ _).l_iSup₂]
- exact iSup_congr fun i => iSup_congr fun hi => (h i hi).2
+ simp only [rootSet_prod p s h0, ← Set.iSup_eq_iUnion, (@gc K _ L _ _).l_iSup₂]
+ exact iSup_congr fun i ↦ iSup_congr fun hi ↦ (h i hi).2
#align intermediate_field.is_splitting_field_supr IntermediateField.isSplittingField_iSup
+end Supremum
+
open Set CompleteLattice
/- Porting note: this was tagged `simp`, but the LHS can be simplified now that the notation
@@ -631,46 +713,25 @@ has been improved. -/
theorem adjoin_simple_le_iff {K : IntermediateField F E} : F⟮α⟯ ≤ K ↔ α ∈ K := by simp
#align intermediate_field.adjoin_simple_le_iff IntermediateField.adjoin_simple_le_iff
+theorem biSup_adjoin_simple : ⨆ x ∈ S, F⟮x⟯ = adjoin F S :=
+ le_antisymm (iSup_le fun _ ↦ iSup_le fun hx ↦ adjoin_simple_le_iff.mpr <| subset_adjoin F S hx) <|
+ adjoin_le_iff.mpr fun x hx ↦ adjoin_simple_le_iff.mp (le_iSup_of_le x (le_iSup_of_le hx le_rfl))
+
/-- Adjoining a single element is compact in the lattice of intermediate fields. -/
theorem adjoin_simple_isCompactElement (x : E) : IsCompactElement F⟮x⟯ := by
- rw [isCompactElement_iff_le_of_directed_sSup_le]
- rintro s ⟨F₀, hF₀⟩ hs hx
- simp only [adjoin_simple_le_iff] at hx ⊢
- let F : IntermediateField F E :=
- { carrier := ⋃ E ∈ s, ↑E
- add_mem' := by
- rintro x₁ x₂ ⟨-, ⟨F₁, rfl⟩, ⟨-, ⟨hF₁, rfl⟩, hx₁⟩⟩ ⟨-, ⟨F₂, rfl⟩, ⟨-, ⟨hF₂, rfl⟩, hx₂⟩⟩
- obtain ⟨F₃, hF₃, h₁₃, h₂₃⟩ := hs F₁ hF₁ F₂ hF₂
- exact mem_iUnion_of_mem F₃ (mem_iUnion_of_mem hF₃ (F₃.add_mem (h₁₃ hx₁) (h₂₃ hx₂)))
- mul_mem' := by
- rintro x₁ x₂ ⟨-, ⟨F₁, rfl⟩, ⟨-, ⟨hF₁, rfl⟩, hx₁⟩⟩ ⟨-, ⟨F₂, rfl⟩, ⟨-, ⟨hF₂, rfl⟩, hx₂⟩⟩
- obtain ⟨F₃, hF₃, h₁₃, h₂₃⟩ := hs F₁ hF₁ F₂ hF₂
- exact mem_iUnion_of_mem F₃ (mem_iUnion_of_mem hF₃ (F₃.mul_mem (h₁₃ hx₁) (h₂₃ hx₂)))
- inv_mem' := by
- rintro x ⟨-, ⟨E, rfl⟩, ⟨-, ⟨hE, rfl⟩, hx⟩⟩
- exact mem_iUnion_of_mem E (mem_iUnion_of_mem hE (E.inv_mem hx))
- algebraMap_mem' := fun x =>
- mem_iUnion_of_mem F₀ (mem_iUnion_of_mem hF₀ (F₀.algebraMap_mem x)) }
--- Porting note: original proof of `key` was
--- `sSup_le fun E1 hE1 => Set.subset_iUnion_of_subset E1 (subset_iUnion _ hE1)`
- have key : sSup s ≤ F := sSup_le fun E1 hE1 => by
- refine' Set.subset_iUnion_of_subset E1 _
- intro x hx
- simpa [hE1] using hx
- obtain ⟨-, ⟨E, rfl⟩, -, ⟨hE, rfl⟩, hx⟩ := key hx
- exact ⟨E, hE, hx⟩
+ simp_rw [isCompactElement_iff_le_of_directed_sSup_le,
+ adjoin_simple_le_iff, sSup_eq_iSup', ← exists_prop]
+ intro s hne hs hx
+ have := hne.to_subtype
+ rwa [← SetLike.mem_coe, coe_iSup_of_directed hs.directed_val, mem_iUnion, Subtype.exists] at hx
#align intermediate_field.adjoin_simple_is_compact_element IntermediateField.adjoin_simple_isCompactElement
-- Porting note: original proof times out.
/-- Adjoining a finite subset is compact in the lattice of intermediate fields. -/
theorem adjoin_finset_isCompactElement (S : Finset E) :
IsCompactElement (adjoin F S : IntermediateField F E) := by
- have key : adjoin F ↑S = ⨆ x ∈ S, F⟮x⟯ := by
--- Porting note: `exact` or `apply` timeout here
- refine' le_antisymm (adjoin_le_iff.mpr fun x hx => SetLike.mem_coe.mpr
- (adjoin_simple_le_iff.mp (le_iSup_of_le x (le_iSup_iff.2 (fun E1 hE1 => hE1 hx)))))
- (iSup_le fun x => iSup_le fun hx => adjoin_simple_le_iff.mpr (subset_adjoin F S hx))
- rw [key, ← Finset.sup_eq_iSup]
+ rw [← biSup_adjoin_simple]
+ simp_rw [Finset.mem_coe, ← Finset.sup_eq_iSup]
exact finset_sup_compact_of_compact S fun x _ => adjoin_simple_isCompactElement x
#align intermediate_field.adjoin_finset_is_compact_element IntermediateField.adjoin_finset_isCompactElement
@@ -684,9 +745,8 @@ instance : IsCompactlyGenerated (IntermediateField F E) :=
⟨fun s =>
⟨(fun x => F⟮x⟯) '' s,
⟨by rintro t ⟨x, _, rfl⟩; exact adjoin_simple_isCompactElement x,
- sSup_image.trans
- (le_antisymm (iSup_le fun i => iSup_le fun hi => adjoin_simple_le_iff.mpr hi) fun x hx =>
- adjoin_simple_le_iff.mp (le_iSup_of_le x (le_iSup_of_le hx le_rfl)))⟩⟩⟩
+ sSup_image.trans <| (biSup_adjoin_simple _).trans <|
+ le_antisymm (adjoin_le_iff.mpr le_rfl) <| subset_adjoin F (s : Set E)⟩⟩⟩
theorem exists_finset_of_mem_iSup {ι : Type*} {f : ι → IntermediateField F E} {x : E}
(hx : x ∈ ⨆ i, f i) : ∃ s : Finset ι, x ∈ ⨆ i ∈ s, f i := by
@@ -699,9 +759,8 @@ theorem exists_finset_of_mem_iSup {ι : Type*} {f : ι → IntermediateField F E
theorem exists_finset_of_mem_supr' {ι : Type*} {f : ι → IntermediateField F E} {x : E}
(hx : x ∈ ⨆ i, f i) : ∃ s : Finset (Σ i, f i), x ∈ ⨆ i ∈ s, F⟮(i.2 : E)⟯ := by
-- Porting note: writing `fun i x h => ...` does not work.
- refine' exists_finset_of_mem_iSup (SetLike.le_def.mp (iSup_le (fun i => _)) hx)
- intro x h
- exact SetLike.le_def.mp (le_iSup_of_le ⟨i, x, h⟩ (by simp)) (mem_adjoin_simple_self F x)
+ refine exists_finset_of_mem_iSup (SetLike.le_def.mp (iSup_le fun i ↦ ?_) hx)
+ exact fun x h ↦ SetLike.le_def.mp (le_iSup_of_le ⟨i, x, h⟩ (by simp)) (mem_adjoin_simple_self F x)
#align intermediate_field.exists_finset_of_mem_supr' IntermediateField.exists_finset_of_mem_supr'
theorem exists_finset_of_mem_supr'' {ι : Type*} {f : ι → IntermediateField F E}
@@ -917,6 +976,9 @@ theorem adjoin.finiteDimensional {x : L} (hx : IsIntegral K x) : FiniteDimension
PowerBasis.finiteDimensional (adjoin.powerBasis hx)
#align intermediate_field.adjoin.finite_dimensional IntermediateField.adjoin.finiteDimensional
+theorem isAlgebraic_adjoin_simple {x : L} (hx : IsIntegral K x) : Algebra.IsAlgebraic K K⟮x⟯ :=
+ have := adjoin.finiteDimensional hx; Algebra.isAlgebraic_of_finite K K⟮x⟯
+
theorem adjoin.finrank {x : L} (hx : IsIntegral K x) :
FiniteDimensional.finrank K K⟮x⟯ = (minpoly K x).natDegree := by
rw [PowerBasis.finrank (adjoin.powerBasis hx : _)]
@@ -977,6 +1039,32 @@ theorem _root_.minpoly.degree_le (x : L) [FiniteDimensional K L] :
degree_le_of_natDegree_le (minpoly.natDegree_le x)
#align minpoly.degree_le minpoly.degree_le
+/-- A compositum of algebraic extensions is algebraic -/
+theorem isAlgebraic_iSup {ι : Type*} {t : ι → IntermediateField K L}
+ (h : ∀ i, Algebra.IsAlgebraic K (t i)) :
+ Algebra.IsAlgebraic K (⨆ i, t i : IntermediateField K L) := by
+ rintro ⟨x, hx⟩
+ obtain ⟨s, hx⟩ := exists_finset_of_mem_supr' hx
+ rw [isAlgebraic_iff, Subtype.coe_mk, ← Subtype.coe_mk (p := (· ∈ _)) x hx, ← isAlgebraic_iff]
+ haveI : ∀ i : Σ i, t i, FiniteDimensional K K⟮(i.2 : L)⟯ := fun ⟨i, x⟩ ↦
+ adjoin.finiteDimensional (isIntegral_iff.1 (isAlgebraic_iff_isIntegral.1 (h i x)))
+ apply Algebra.isAlgebraic_of_finite
+#align intermediate_field.is_algebraic_supr IntermediateField.isAlgebraic_iSup
+
+theorem isAlgebraic_adjoin {S : Set L} (hS : ∀ x ∈ S, IsIntegral K x) :
+ Algebra.IsAlgebraic K (adjoin K S) := by
+ rw [← biSup_adjoin_simple, ← iSup_subtype'']
+ exact isAlgebraic_iSup fun x ↦ isAlgebraic_adjoin_simple (hS x x.2)
+
+/-- If `L / K` is a field extension, `S` is a finite subset of `L`, such that every element of `S`
+is integral (= algebraic) over `K`, then `K(S) / K` is a finite extension.
+A direct corollary of `finiteDimensional_iSup_of_finite`. -/
+theorem finiteDimensional_adjoin {S : Set L} [Finite S] (hS : ∀ x ∈ S, IsIntegral K x) :
+ FiniteDimensional K (adjoin K S) := by
+ rw [← biSup_adjoin_simple, ← iSup_subtype'']
+ haveI (x : S) := adjoin.finiteDimensional (hS x.1 x.2)
+ exact finiteDimensional_iSup_of_finite
+
end PowerBasis
/-- Algebra homomorphism `F⟮α⟯ →ₐ[F] K` are in bijection with the set of roots
@@ -1042,7 +1130,7 @@ theorem fg_of_noetherian (S : IntermediateField F E) [IsNoetherian F E] : S.FG :
theorem induction_on_adjoin_finset (S : Finset E) (P : IntermediateField F E → Prop) (base : P ⊥)
(ih : ∀ (K : IntermediateField F E), ∀ x ∈ S, P K → P (K⟮x⟯.restrictScalars F)) :
- P (adjoin F ↑S) := by
+ P (adjoin F S) := by
refine' Finset.induction_on' S _ (fun ha _ _ h => _)
· simp [base]
· rw [Finset.coe_insert, Set.insert_eq, Set.union_comm, ← adjoin_adjoin_left]
@@ -1070,309 +1158,132 @@ section AlgHomMkAdjoinSplits
variable (F E K : Type*) [Field F] [Field E] [Field K] [Algebra F E] [Algebra F K] {S : Set E}
/-- Lifts `L → K` of `F → K` -/
-def Lifts :=
- Σ L : IntermediateField F E, L →ₐ[F] K
+structure Lifts where
+ /-- The domain of a lift. -/
+ carrier : IntermediateField F E
+ /-- The lifted RingHom, expressed as an AlgHom. -/
+ emb : carrier →ₐ[F] K
#align intermediate_field.lifts IntermediateField.Lifts
variable {F E K}
instance : PartialOrder (Lifts F E K) where
- le x y := x.1 ≤ y.1 ∧ ∀ (s : x.1) (t : y.1), (s : E) = t → x.2 s = y.2 t
- le_refl x := ⟨le_refl x.1, fun s t hst => congr_arg x.2 (Subtype.ext hst)⟩
- le_trans x y z hxy hyz :=
- ⟨le_trans hxy.1 hyz.1, fun s u hsu =>
- Eq.trans (hxy.2 s ⟨s, hxy.1 s.mem⟩ rfl) (hyz.2 ⟨s, hxy.1 s.mem⟩ u hsu)⟩
+ le L₁ L₂ := ∃ h : L₁.carrier ≤ L₂.carrier, ∀ x, L₂.emb (inclusion h x) = L₁.emb x
+ le_refl L := ⟨le_rfl, by simp⟩
+ le_trans L₁ L₂ L₃ := by
+ rintro ⟨h₁₂, h₁₂'⟩ ⟨h₂₃, h₂₃'⟩
+ refine ⟨h₁₂.trans h₂₃, fun _ ↦ ?_⟩
+ rw [← inclusion_inclusion h₁₂ h₂₃, h₂₃', h₁₂']
le_antisymm := by
- rintro ⟨x1, x2⟩ ⟨y1, y2⟩ ⟨hxy1, hxy2⟩ ⟨hyx1, hyx2⟩
- obtain rfl : x1 = y1 := le_antisymm hxy1 hyx1
+ rintro ⟨L₁, e₁⟩ ⟨L₂, e₂⟩ ⟨h₁₂, h₁₂'⟩ ⟨h₂₁, h₂₁'⟩
+ obtain rfl : L₁ = L₂ := h₁₂.antisymm h₂₁
congr
- exact AlgHom.ext fun s => hxy2 s s rfl
+ exact AlgHom.ext h₂₁'
noncomputable instance : OrderBot (Lifts F E K) where
- bot := ⟨⊥, (Algebra.ofId F K).comp (botEquiv F E).toAlgHom⟩
- bot_le x :=
- ⟨bot_le, fun s t hst => by
- cases' IntermediateField.mem_bot.mp s.mem with u hu
- rw [show s = (algebraMap F _) u from Subtype.ext hu.symm, AlgHom.commutes]
- rw [show t = (algebraMap F _) u from Subtype.ext (Eq.trans hu hst).symm, AlgHom.commutes]⟩
+ bot := ⟨⊥, (Algebra.ofId F K).comp (botEquiv F E)⟩
+ bot_le L := ⟨bot_le, fun x ↦ by
+ obtain ⟨x, rfl⟩ := (botEquiv F E).symm.surjective x
+ simp_rw [AlgHom.comp_apply, AlgHom.coe_coe, AlgEquiv.apply_symm_apply]
+ exact L.emb.commutes x⟩
noncomputable instance : Inhabited (Lifts F E K) :=
⟨⊥⟩
-theorem Lifts.eq_of_le {x y : Lifts F E K} (hxy : x ≤ y) (s : x.1) : x.2 s = y.2 ⟨s, hxy.1 s.mem⟩ :=
- hxy.2 s ⟨s, hxy.1 s.mem⟩ rfl
-#align intermediate_field.lifts.eq_of_le IntermediateField.Lifts.eq_of_le
-
-theorem Lifts.exists_max_two {c : Set (Lifts F E K)} {x y : Lifts F E K} (hc : IsChain (· ≤ ·) c)
- (hx : x ∈ insert ⊥ c) (hy : y ∈ insert ⊥ c) :
- ∃ z : Lifts F E K, z ∈ insert ⊥ c ∧ x ≤ z ∧ y ≤ z := by
- cases' (hc.insert fun _ _ _ => Or.inl bot_le).total hx hy with hxy hyx
- · exact ⟨y, hy, hxy, le_refl y⟩
- · exact ⟨x, hx, le_refl x, hyx⟩
-#align intermediate_field.lifts.exists_max_two IntermediateField.Lifts.exists_max_two
-
-theorem Lifts.exists_max_three {c : Set (Lifts F E K)} {x y z : Lifts F E K}
- (hc : IsChain (· ≤ ·) c) (hx : x ∈ insert ⊥ c) (hy : y ∈ insert ⊥ c)
- (hz : z ∈ insert ⊥ c) :
- ∃ w : Lifts F E K, w ∈ insert ⊥ c ∧ x ≤ w ∧ y ≤ w ∧ z ≤ w := by
- obtain ⟨v, hv, hxv, hyv⟩ := Lifts.exists_max_two hc hx hy
- obtain ⟨w, hw, hzw, hvw⟩ := Lifts.exists_max_two hc hz hv
- exact ⟨w, hw, le_trans hxv hvw, le_trans hyv hvw, hzw⟩
-#align intermediate_field.lifts.exists_max_three IntermediateField.Lifts.exists_max_three
-
-/-- An upper bound on a chain of lifts -/
-def Lifts.upperBoundIntermediateField {c : Set (Lifts F E K)} (hc : IsChain (· ≤ ·) c) :
- IntermediateField F E where
- carrier s := ∃ x : Lifts F E K, x ∈ insert ⊥ c ∧ (s ∈ x.1 : Prop)
- zero_mem' := ⟨⊥, Set.mem_insert ⊥ c, zero_mem ⊥⟩
- one_mem' := ⟨⊥, Set.mem_insert ⊥ c, one_mem ⊥⟩
- inv_mem' := by rintro _ ⟨x, y, h⟩; exact ⟨x, ⟨y, x.1.inv_mem h⟩⟩
- add_mem' := by
- rintro _ _ ⟨x, hx, ha⟩ ⟨y, hy, hb⟩
- obtain ⟨z, hz, hxz, hyz⟩ := Lifts.exists_max_two hc hx hy
- exact ⟨z, hz, z.1.add_mem (hxz.1 ha) (hyz.1 hb)⟩
- mul_mem' := by
- rintro _ _ ⟨x, hx, ha⟩ ⟨y, hy, hb⟩
- obtain ⟨z, hz, hxz, hyz⟩ := Lifts.exists_max_two hc hx hy
- exact ⟨z, hz, z.1.mul_mem (hxz.1 ha) (hyz.1 hb)⟩
- algebraMap_mem' s := ⟨⊥, Set.mem_insert ⊥ c, algebraMap_mem ⊥ s⟩
-#align intermediate_field.lifts.upper_bound_intermediate_field IntermediateField.Lifts.upperBoundIntermediateField
-
-/-- The lift on the upper bound on a chain of lifts -/
-noncomputable def Lifts.upperBoundAlgHom {c : Set (Lifts F E K)} (hc : IsChain (· ≤ ·) c) :
- Lifts.upperBoundIntermediateField hc →ₐ[F] K where
- toFun s := (Classical.choose s.mem).2 ⟨s, (Classical.choose_spec s.mem).2⟩
- map_zero' := AlgHom.map_zero _
- map_one' := AlgHom.map_one _
- map_add' s t := by
- obtain ⟨w, _, hxw, hyw, hzw⟩ :=
- Lifts.exists_max_three hc (Classical.choose_spec s.mem).1 (Classical.choose_spec t.mem).1
- (Classical.choose_spec (s + t).mem).1
-
- simp only [Subsemiring.coe_add, Subalgebra.coe_toSubsemiring, coe_toSubalgebra,
- Lifts.eq_of_le hzw, Lifts.eq_of_le hxw, Lifts.eq_of_le hyw, ← w.2.map_add,
- AddMemClass.mk_add_mk]
- map_mul' s t := by
- obtain ⟨w, _, hxw, hyw, hzw⟩ :=
- Lifts.exists_max_three hc (Classical.choose_spec s.mem).1 (Classical.choose_spec t.mem).1
- (Classical.choose_spec (s * t).mem).1
- simp only [Submonoid.coe_mul, Subsemiring.coe_toSubmonoid, Subalgebra.coe_toSubsemiring,
- coe_toSubalgebra, Lifts.eq_of_le hzw, Lifts.eq_of_le hxw, Lifts.eq_of_le hyw, ← w.2.map_mul,
- Submonoid.mk_mul_mk]
- commutes' _ := AlgHom.commutes _ _
-#align intermediate_field.lifts.upper_bound_alg_hom IntermediateField.Lifts.upperBoundAlgHom
-
-/-- An upper bound on a chain of lifts -/
-noncomputable def Lifts.upperBound {c : Set (Lifts F E K)} (hc : IsChain (· ≤ ·) c) : Lifts F E K :=
- ⟨Lifts.upperBoundIntermediateField hc, Lifts.upperBoundAlgHom hc⟩
-#align intermediate_field.lifts.upper_bound IntermediateField.Lifts.upperBound
-
+/-- A chain of lifts has an upper bound. -/
theorem Lifts.exists_upper_bound (c : Set (Lifts F E K)) (hc : IsChain (· ≤ ·) c) :
- ∃ ub, ∀ a ∈ c, a ≤ ub :=
- ⟨Lifts.upperBound hc, by
- intro x hx
- constructor
- · exact fun s hs => ⟨x, Set.mem_insert_of_mem ⊥ hx, hs⟩
- · intro s t hst
- change x.2 s = (Classical.choose t.mem).2 ⟨t, (Classical.choose_spec t.mem).2⟩
- obtain ⟨z, _, hxz, hyz⟩ :=
- Lifts.exists_max_two hc (Set.mem_insert_of_mem ⊥ hx) (Classical.choose_spec t.mem).1
- rw [Lifts.eq_of_le hxz, Lifts.eq_of_le hyz]
- exact congr_arg z.2 (Subtype.ext hst)⟩
+ ∃ ub, ∀ a ∈ c, a ≤ ub := by
+ let t (i : ↑(insert ⊥ c)) := i.val.carrier
+ let t' (i) := (t i).toSubalgebra
+ have hc := hc.insert fun _ _ _ ↦ .inl bot_le
+ have dir : Directed (· ≤ ·) t := hc.directedOn.directed_val.mono_comp _ fun _ _ h ↦ h.1
+ refine ⟨⟨iSup t, (Subalgebra.iSupLift t' dir (fun i ↦ i.val.emb) (fun i j h ↦ ?_) _ rfl).comp
+ (Subalgebra.equivOfEq _ _ <| toSubalgebra_iSup_of_directed dir)⟩,
+ fun L hL ↦ have hL := Set.mem_insert_of_mem ⊥ hL; ⟨le_iSup t ⟨L, hL⟩, fun x ↦ ?_⟩⟩
+ · refine AlgHom.ext fun x ↦ (hc.total i.2 j.2).elim (fun hij ↦ (hij.snd x).symm) fun hji ↦ ?_
+ erw [AlgHom.comp_apply, ← hji.snd (Subalgebra.inclusion h x),
+ inclusion_inclusion, inclusion_self, AlgHom.id_apply x]
+ · dsimp only [AlgHom.comp_apply]
+ exact Subalgebra.iSupLift_inclusion (K := t') (i := ⟨L, hL⟩) x (le_iSup t' ⟨L, hL⟩)
#align intermediate_field.lifts.exists_upper_bound IntermediateField.Lifts.exists_upper_bound
--- Porting note: instance `alg` added by hand. The proof is very slow.
-/-- Extend a lift `x : Lifts F E K` to an element `s : E` whose conjugates are all in `K` -/
-noncomputable def Lifts.liftOfSplits (x : Lifts F E K) {s : E} (h1 : IsIntegral F s)
- (h2 : (minpoly F s).Splits (algebraMap F K)) : Lifts F E K :=
- let h3 : IsIntegral x.1 s := isIntegral_of_isScalarTower h1
- let key : (minpoly x.1 s).Splits x.2.toRingHom :=
- splits_of_splits_of_dvd _ (map_ne_zero (minpoly.ne_zero h1))
- ((splits_map_iff _ _).mpr (by convert h2; exact RingHom.ext fun y => x.2.commutes y))
- (minpoly.dvd_map_of_isScalarTower _ _ _)
- letI : Algebra {y // y ∈ x.fst} {y // y ∈ restrictScalars F {z // z ∈ x.fst}⟮s⟯} :=
- {z // z ∈ x.fst}⟮s⟯.toSubalgebra.algebra
- letI := x.2.toRingHom.toAlgebra
- ⟨x.1⟮s⟯.restrictScalars F,
- (@algHomEquivSigma F x.1 (x.1⟮s⟯.restrictScalars F) K _ _ _ _ _ _ _
- (IntermediateField.algebra x.1⟮s⟯) (IsScalarTower.of_algebraMap_eq fun _ => rfl)).invFun
- ⟨x.2,
- (@algHomAdjoinIntegralEquiv x.1 _ E _ _ s K _ _ h3).invFun
- ⟨rootOfSplits x.2.toRingHom key (ne_of_gt (minpoly.degree_pos h3)), by
- rw [mem_aroots, and_iff_right (minpoly.ne_zero h3)]
- exact map_rootOfSplits x.2.toRingHom key (ne_of_gt (minpoly.degree_pos h3))⟩⟩⟩
-#align intermediate_field.lifts.lift_of_splits IntermediateField.Lifts.liftOfSplits
-
--- Porting note: instance `alg` added by hand.
--- Porting note: Lean3 is able to guess `φ`
-theorem Lifts.le_lifts_of_splits (x : Lifts F E K) {s : E} (h1 : IsIntegral F s)
- (h2 : (minpoly F s).Splits (algebraMap F K)) : x ≤ x.liftOfSplits h1 h2 :=
- ⟨fun z hz => algebraMap_mem x.1⟮s⟯ ⟨z, hz⟩, fun t u htu =>
- Eq.symm
- (by
- let alg : Algebra x.1 x.1⟮s⟯ := x.1⟮s⟯.toSubalgebra.algebra
- rw [← show algebraMap x.1 x.1⟮s⟯ t = u from Subtype.ext htu]
- letI : Algebra x.1 K := x.2.toRingHom.toAlgebra
- let I1 : IsIntegral x.1 s := isIntegral_of_isScalarTower h1
- let key : (minpoly x.1 s).Splits x.2.toRingHom :=
- splits_of_splits_of_dvd _ (map_ne_zero (minpoly.ne_zero h1))
- ((splits_map_iff _ _).mpr (by convert h2; exact RingHom.ext fun y => x.2.commutes y))
- (minpoly.dvd_map_of_isScalarTower _ _ _)
- have I2 := (ne_of_gt (minpoly.degree_pos I1))
- have I3 : rootOfSplits (AlgHom.toRingHom x.2) key (ne_of_gt (minpoly.degree_pos I1)) ∈
- (minpoly x.1 s).aroots K := by
- rw [mem_aroots, and_iff_right (minpoly.ne_zero I1)]
- exact map_rootOfSplits x.2.toRingHom key (ne_of_gt (minpoly.degree_pos I1))
- let φ : x.1⟮s⟯ →ₐ[x.1] K := ((algHomAdjoinIntegralEquiv x.1 I1).invFun
- ⟨rootOfSplits (AlgHom.toRingHom x.2) key I2, I3⟩)
-
- exact AlgHom.commutes φ t)⟩
-#align intermediate_field.lifts.le_lifts_of_splits IntermediateField.Lifts.le_lifts_of_splits
-
-theorem Lifts.mem_lifts_of_splits (x : Lifts F E K) {s : E} (h1 : IsIntegral F s)
- (h2 : (minpoly F s).Splits (algebraMap F K)) : s ∈ (x.liftOfSplits h1 h2).1 :=
- mem_adjoin_simple_self x.1 s
-#align intermediate_field.lifts.mem_lifts_of_splits IntermediateField.Lifts.mem_lifts_of_splits
-
+/-- Given an element `s : E` whose conjugates are all in `K`, any lift can be extended to one
+ whose carrier contains `s`. -/
theorem Lifts.exists_lift_of_splits (x : Lifts F E K) {s : E} (h1 : IsIntegral F s)
- (h2 : (minpoly F s).Splits (algebraMap F K)) : ∃ y, x ≤ y ∧ s ∈ y.1 :=
- ⟨x.liftOfSplits h1 h2, x.le_lifts_of_splits h1 h2, x.mem_lifts_of_splits h1 h2⟩
+ (h2 : (minpoly F s).Splits (algebraMap F K)) : ∃ y, x ≤ y ∧ s ∈ y.carrier :=
+ have I1 : IsIntegral x.carrier s := isIntegral_of_isScalarTower h1
+ have I2 := (minpoly.degree_pos I1).ne'
+ have key : (minpoly x.carrier s).Splits x.emb.toRingHom :=
+ splits_of_splits_of_dvd _ (map_ne_zero (minpoly.ne_zero h1))
+ ((splits_map_iff _ _).2 (x.emb.comp_algebraMap ▸ h2)) (minpoly.dvd_map_of_isScalarTower _ _ _)
+ letI : Algebra x.carrier K := x.emb.toRingHom.toAlgebra
+ let carrier := x.carrier⟮s⟯.restrictScalars F
+ letI : Algebra x.carrier carrier := x.carrier⟮s⟯.toSubalgebra.algebra
+ let φ : carrier →ₐ[x.carrier] K := ((algHomAdjoinIntegralEquiv x.carrier I1).symm
+ ⟨rootOfSplits x.emb.toRingHom key I2, by
+ rw [mem_aroots, and_iff_right (minpoly.ne_zero I1)]
+ exact map_rootOfSplits x.emb.toRingHom key I2⟩)
+ ⟨⟨carrier, (@algHomEquivSigma F x.carrier carrier K _ _ _ _ _ _ _ _
+ (IsScalarTower.of_algebraMap_eq fun _ ↦ rfl)).symm ⟨x.emb, φ⟩⟩,
+ ⟨fun z hz ↦ algebraMap_mem x.carrier⟮s⟯ ⟨z, hz⟩, φ.commutes⟩,
+ mem_adjoin_simple_self x.carrier s⟩
#align intermediate_field.lifts.exists_lift_of_splits IntermediateField.Lifts.exists_lift_of_splits
-theorem algHom_mk_adjoin_splits
- (hK : ∀ s ∈ S, IsIntegral F (s : E) ∧ (minpoly F s).Splits (algebraMap F K)) :
- Nonempty (adjoin F S →ₐ[F] K) := by
- obtain ⟨x, hx⟩ : ∃ m : Lifts F E K, ∀ a, m ≤ a → a = m :=
- zorn_partialOrder Lifts.exists_upper_bound
- refine'
- ⟨{ toFun := (fun s => x.2 ⟨s, adjoin_le_iff.mpr (fun s hs => _) s.mem⟩)
- map_one' := x.2.map_one
- map_mul' := (fun s t => x.2.map_mul ⟨s, _⟩ ⟨t, _⟩)
- map_zero' := x.2.map_zero
- map_add' := (fun s t => x.2.map_add ⟨s, _⟩ ⟨t, _⟩)
- commutes' := x.2.commutes }⟩
- rcases x.exists_lift_of_splits (hK s hs).1 (hK s hs).2 with ⟨y, h1, h2⟩
- rwa [hx y h1] at h2
-#align intermediate_field.alg_hom_mk_adjoin_splits IntermediateField.algHom_mk_adjoin_splits
-
-theorem algHom_mk_adjoin_splits' (hS : adjoin F S = ⊤)
- (hK : ∀ x ∈ S, IsIntegral F (x : E) ∧ (minpoly F x).Splits (algebraMap F K)) :
- Nonempty (E →ₐ[F] K) := by
- cases' algHom_mk_adjoin_splits hK with ϕ
- rw [hS] at ϕ
- exact ⟨ϕ.comp topEquiv.symm.toAlgHom⟩
-#align intermediate_field.alg_hom_mk_adjoin_splits' IntermediateField.algHom_mk_adjoin_splits'
+variable (hK : ∀ s ∈ S, IsIntegral F s ∧ (minpoly F s).Splits (algebraMap F K))
+ (hK' : ∀ s : E, IsIntegral F s ∧ (minpoly F s).Splits (algebraMap F K))
+ {L : IntermediateField F E} (f : L →ₐ[F] K) (hL : L ≤ adjoin F S)
-end AlgHomMkAdjoinSplits
+-- The following uses the hypothesis `hK`.
-section Supremum
+theorem exists_algHom_adjoin_of_splits : ∃ φ : adjoin F S →ₐ[F] K, φ.comp (inclusion hL) = f := by
+ obtain ⟨φ, hfφ, hφ⟩ := zorn_nonempty_Ici₀ _
+ (fun c _ hc _ _ ↦ Lifts.exists_upper_bound c hc) ⟨L, f⟩ le_rfl
+ refine ⟨φ.emb.comp (inclusion <| adjoin_le_iff.mpr fun s hs ↦ ?_), ?_⟩
+ · rcases φ.exists_lift_of_splits (hK s hs).1 (hK s hs).2 with ⟨y, h1, h2⟩
+ exact (hφ y h1).1 h2
+ · ext; apply hfφ.2
-variable {K L : Type*} [Field K] [Field L] [Algebra K L] (E1 E2 : IntermediateField K L)
+theorem nonempty_algHom_adjoin_of_splits : Nonempty (adjoin F S →ₐ[F] K) :=
+ have ⟨φ, _⟩ := exists_algHom_adjoin_of_splits hK (⊥ : Lifts F E K).emb bot_le; ⟨φ⟩
+#align intermediate_field.alg_hom_mk_adjoin_splits IntermediateField.nonempty_algHom_adjoin_of_splits
-theorem le_sup_toSubalgebra : E1.toSubalgebra ⊔ E2.toSubalgebra ≤ (E1 ⊔ E2).toSubalgebra :=
- sup_le (show E1 ≤ E1 ⊔ E2 from le_sup_left) (show E2 ≤ E1 ⊔ E2 from le_sup_right)
-#align intermediate_field.le_sup_to_subalgebra IntermediateField.le_sup_toSubalgebra
+variable (hS : adjoin F S = ⊤)
-theorem sup_toSubalgebra [h1 : FiniteDimensional K E1] [h2 : FiniteDimensional K E2] :
- (E1 ⊔ E2).toSubalgebra = E1.toSubalgebra ⊔ E2.toSubalgebra := by
- let S1 := E1.toSubalgebra
- let S2 := E2.toSubalgebra
- refine'
- le_antisymm
- (show _ ≤ (S1 ⊔ S2).toIntermediateField _ from
- sup_le (show S1 ≤ _ from le_sup_left) (show S2 ≤ _ from le_sup_right))
- (le_sup_toSubalgebra E1 E2)
- suffices IsField (S1 ⊔ S2 : Subalgebra K L) by
- intro x hx
- by_cases hx' : (⟨x, hx⟩ : (S1 ⊔ S2 : Subalgebra K L)) = 0
- · rw [← Subtype.coe_mk x, hx', Subalgebra.coe_zero, inv_zero]
- exact (S1 ⊔ S2).zero_mem
- · obtain ⟨y, h⟩ := this.mul_inv_cancel hx'
- exact (congr_arg (· ∈ S1 ⊔ S2) <| eq_inv_of_mul_eq_one_right <| Subtype.ext_iff.mp h).mp y.2
- exact
- isField_of_isIntegral_of_isField'
- (isIntegral_sup.mpr ⟨Algebra.isIntegral_of_finite K E1, Algebra.isIntegral_of_finite K E2⟩)
- (Field.toIsField K)
-#align intermediate_field.sup_to_subalgebra IntermediateField.sup_toSubalgebra
+theorem exists_algHom_of_adjoin_splits : ∃ φ : E →ₐ[F] K, φ.comp L.val = f :=
+ have ⟨φ, hφ⟩ := exists_algHom_adjoin_of_splits hK f (hS.symm ▸ le_top)
+ ⟨φ.comp ((equivOfEq hS).trans topEquiv).symm.toAlgHom, hφ⟩
-instance finiteDimensional_sup [h1 : FiniteDimensional K E1] [h2 : FiniteDimensional K E2] :
- FiniteDimensional K (E1 ⊔ E2 : IntermediateField K L) := by
- let g := Algebra.TensorProduct.productMap E1.val E2.val
- suffices g.range = (E1 ⊔ E2).toSubalgebra by
- have h : FiniteDimensional K (Subalgebra.toSubmodule g.range) :=
- g.toLinearMap.finiteDimensional_range
- rwa [this] at h
- rw [Algebra.TensorProduct.productMap_range, E1.range_val, E2.range_val, sup_toSubalgebra]
-#align intermediate_field.finite_dimensional_sup IntermediateField.finiteDimensional_sup
+theorem nonempty_algHom_of_adjoin_splits : Nonempty (E →ₐ[F] K) :=
+ have ⟨φ, _⟩ := exists_algHom_of_adjoin_splits hK (⊥ : Lifts F E K).emb hS; ⟨φ⟩
+#align intermediate_field.alg_hom_mk_adjoin_splits' IntermediateField.nonempty_algHom_of_adjoin_splits
-instance finiteDimensional_iSup_of_finite {ι : Type*} {t : ι → IntermediateField K L}
- [h : Finite ι] [∀ i, FiniteDimensional K (t i)] :
- FiniteDimensional K (⨆ i, t i : IntermediateField K L) := by
- rw [← iSup_univ]
- let P : Set ι → Prop := fun s => FiniteDimensional K (⨆ i ∈ s, t i : IntermediateField K L)
- change P Set.univ
- apply Set.Finite.induction_on
- all_goals dsimp only
- · exact Set.finite_univ
- · rw [iSup_emptyset]
- exact (botEquiv K L).symm.toLinearEquiv.finiteDimensional
- · intro _ s _ _ hs
- rw [iSup_insert]
- exact IntermediateField.finiteDimensional_sup _ _
-#align intermediate_field.finite_dimensional_supr_of_finite IntermediateField.finiteDimensional_iSup_of_finite
+variable {x : E} (hx : x ∈ adjoin F S) {y : K} (hy : aeval y (minpoly F x) = 0)
-/-- If `L / K` is an algebraic extension, then for any finite subset `S` of `L`, `K(S) / K` is a
-finite extension. A direct corollary of `finiteDimensional_iSup_of_finite`. -/
-theorem finiteDimensional_adjoin_of_finite_of_isAlgebraic
- (halg : Algebra.IsAlgebraic K L) (S : Set L) [Finite S] :
- FiniteDimensional K (adjoin K S) := by
- let t : S → IntermediateField K L := fun x ↦ K⟮x.1⟯
- have h : ∀ x : S, FiniteDimensional K (t x) := fun x ↦
- adjoin.finiteDimensional <| isAlgebraic_iff_isIntegral.1 (halg x.1)
- have hfin := finiteDimensional_iSup_of_finite (t := t)
- have := (gc (F := K) (E := L)).l_iSup (f := fun (x : S) ↦ {x.1})
- rw [Set.iSup_eq_iUnion, Set.iUnion_singleton_eq_range, Subtype.range_coe_subtype,
- Set.setOf_mem_eq] at this
- rwa [← this] at hfin
-
-instance finiteDimensional_iSup_of_finset {ι : Type*} {f : ι → IntermediateField K L}
- /-Porting note: changed `h` from `∀ i ∈ s, FiniteDimensional K (f i)` because this caused an
- error. See `finiteDimensional_iSup_of_finset'` for a stronger version, that was the one
- used in mathlib3.-/
- {s : Finset ι} [h : ∀ i, FiniteDimensional K (f i)] :
- FiniteDimensional K (⨆ i ∈ s, f i : IntermediateField K L) := by
- haveI : ∀ i : { i // i ∈ s }, FiniteDimensional K (f i) := fun i => h i
- have : ⨆ i ∈ s, f i = ⨆ i : { i // i ∈ s }, f i :=
- le_antisymm (iSup_le fun i => iSup_le fun h => le_iSup (fun i : { i // i ∈ s } => f i) ⟨i, h⟩)
- (iSup_le fun i => le_iSup_of_le i (le_iSup_of_le i.2 le_rfl))
- exact this.symm ▸ IntermediateField.finiteDimensional_iSup_of_finite
-#align intermediate_field.finite_dimensional_supr_of_finset IntermediateField.finiteDimensional_iSup_of_finset
+theorem exists_algHom_adjoin_of_splits_of_aeval : ∃ φ : adjoin F S →ₐ[F] K, φ ⟨x, hx⟩ = y := by
+ have ix := isAlgebraic_adjoin (fun s hs ↦ (hK s hs).1) ⟨x, hx⟩
+ rw [isAlgebraic_iff_isIntegral, isIntegral_iff] at ix
+ obtain ⟨φ, hφ⟩ := exists_algHom_adjoin_of_splits hK ((algHomAdjoinIntegralEquiv F ix).symm
+ ⟨y, mem_aroots.mpr ⟨minpoly.ne_zero ix, hy⟩⟩) (adjoin_simple_le_iff.mpr hx)
+ exact ⟨φ, (FunLike.congr_fun hφ <| AdjoinSimple.gen F x).trans <|
+ algHomAdjoinIntegralEquiv_symm_apply_gen F ix _⟩
-theorem finiteDimensional_iSup_of_finset' {ι : Type*} {f : ι → IntermediateField K L}
- /-Porting note: this was the mathlib3 version. Using `[h : ...]`, as in mathlib3, causes the
- error "invalid parametric local instance".-/
- {s : Finset ι} (h : ∀ i, i ∈ s → FiniteDimensional K (f i)) :
- FiniteDimensional K (⨆ i ∈ s, f i : IntermediateField K L) := by
- haveI : ∀ i : { i // i ∈ s }, FiniteDimensional K (f i) := fun i => h i i.2
- have : ⨆ i ∈ s, f i = ⨆ i : { i // i ∈ s }, f i :=
- le_antisymm (iSup_le fun i => iSup_le fun h => le_iSup (fun i : { i // i ∈ s } => f i) ⟨i, h⟩)
- (iSup_le fun i => le_iSup_of_le i (le_iSup_of_le i.2 le_rfl))
- exact this.symm ▸ IntermediateField.finiteDimensional_iSup_of_finite
+theorem exists_algHom_of_adjoin_splits_of_aeval : ∃ φ : E →ₐ[F] K, φ x = y :=
+ have ⟨φ, hφ⟩ := exists_algHom_adjoin_of_splits_of_aeval hK (hS ▸ mem_top) hy
+ ⟨φ.comp ((equivOfEq hS).trans topEquiv).symm.toAlgHom, hφ⟩
-/-- A compositum of algebraic extensions is algebraic -/
-theorem isAlgebraic_iSup {ι : Type*} {f : ι → IntermediateField K L}
- (h : ∀ i, Algebra.IsAlgebraic K (f i)) :
- Algebra.IsAlgebraic K (⨆ i, f i : IntermediateField K L) := by
- rintro ⟨x, hx⟩
- obtain ⟨s, hx⟩ := exists_finset_of_mem_supr' hx
- rw [isAlgebraic_iff, Subtype.coe_mk, ← Subtype.coe_mk (p := fun x => x ∈ _) x hx,
- ← isAlgebraic_iff]
- haveI : ∀ i : Σ i, f i, FiniteDimensional K K⟮(i.2 : L)⟯ := fun ⟨i, x⟩ =>
- adjoin.finiteDimensional (isIntegral_iff.1 (isAlgebraic_iff_isIntegral.1 (h i x)))
- apply Algebra.isAlgebraic_of_finite
-#align intermediate_field.is_algebraic_supr IntermediateField.isAlgebraic_iSup
+/- The following uses the hypothesis
+ (hK' : ∀ s : E, IsIntegral F s ∧ (minpoly F s).Splits (algebraMap F K)) -/
-end Supremum
+theorem exists_algHom_of_splits : ∃ φ : E →ₐ[F] K, φ.comp L.val = f :=
+ exists_algHom_of_adjoin_splits (fun x _ ↦ hK' x) f (adjoin_univ F E)
+
+theorem nonempty_algHom_of_splits : Nonempty (E →ₐ[F] K) :=
+ nonempty_algHom_of_adjoin_splits (fun x _ ↦ hK' x) (adjoin_univ F E)
+
+theorem exists_algHom_of_splits_of_aeval : ∃ φ : E →ₐ[F] K, φ x = y :=
+ exists_algHom_of_adjoin_splits_of_aeval (fun x _ ↦ hK' x) (adjoin_univ F E) hy
+
+end AlgHomMkAdjoinSplits
end IntermediateField
feat(FieldTheory/PrimitiveElement): Steinitz Theorem (#7788)
Added Field.exists_primitive_element_iff_finite_intermediateField: a finite extension E / F has a primitive element if and only if the intermediate fields between E / F are finitely many.
Also known as Steinitz Theorem https://en.wikipedia.org/wiki/Primitive_element_theorem#The_theorems.
Co-authored-by: Junyan Xu <junyanxu.math@gmail.com> Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Junyan Xu <junyanxumath@gmail.com> Co-authored-by: Mario Carneiro <di.gama@gmail.com>
Diff
@@ -48,6 +48,23 @@ def adjoin : IntermediateField F E :=
algebraMap_mem' := fun x => Subfield.subset_closure (Or.inl (Set.mem_range_self x)) }
#align intermediate_field.adjoin IntermediateField.adjoin
+variable {S}
+
+theorem mem_adjoin_iff (x : E) :
+ x ∈ adjoin F S ↔ ∃ r s : MvPolynomial S F,
+ x = MvPolynomial.aeval Subtype.val r / MvPolynomial.aeval Subtype.val s := by
+ simp only [adjoin, mem_mk, Subring.mem_toSubsemiring, Subfield.mem_toSubring,
+ Subfield.mem_closure_iff, ← Algebra.adjoin_eq_ring_closure, Subalgebra.mem_toSubring,
+ Algebra.adjoin_eq_range, AlgHom.mem_range, exists_exists_eq_and]
+ tauto
+
+theorem mem_adjoin_simple_iff {α : E} (x : E) :
+ x ∈ adjoin F {α} ↔ ∃ r s : F[X], x = aeval α r / aeval α s := by
+ simp only [adjoin, mem_mk, Subring.mem_toSubsemiring, Subfield.mem_toSubring,
+ Subfield.mem_closure_iff, ← Algebra.adjoin_eq_ring_closure, Subalgebra.mem_toSubring,
+ Algebra.adjoin_singleton_eq_range_aeval, AlgHom.mem_range, exists_exists_eq_and]
+ tauto
+
end AdjoinDef
section Lattice
@@ -423,6 +440,31 @@ theorem adjoin_map {E' : Type*} [Field E'] [Algebra F E'] (f : E →ₐ[F] E') :
rfl
#align intermediate_field.adjoin_map IntermediateField.adjoin_map
+@[simp]
+theorem lift_adjoin (K : IntermediateField F E) (S : Set K) :
+ lift (adjoin F S) = adjoin F (Subtype.val '' S) :=
+ adjoin_map _ _ _
+
+theorem lift_adjoin_simple (K : IntermediateField F E) (α : K) :
+ lift (adjoin F {α}) = adjoin F {α.1} := by
+ simp only [lift_adjoin, Set.image_singleton]
+
+@[simp]
+theorem lift_bot (K : IntermediateField F E) :
+ lift (F := K) ⊥ = ⊥ := map_bot _
+
+@[simp]
+theorem lift_top (K : IntermediateField F E) :
+ lift (F := K) ⊤ = K := by rw [lift, ←AlgHom.fieldRange_eq_map, fieldRange_val]
+
+@[simp]
+theorem adjoin_self (K : IntermediateField F E) :
+ adjoin F K = K := le_antisymm (adjoin_le_iff.2 fun _ ↦ id) (subset_adjoin F _)
+
+theorem restrictScalars_adjoin (K : IntermediateField F E) (S : Set E) :
+ restrictScalars F (adjoin K S) = adjoin F (K ∪ S) := by
+ rw [← adjoin_self _ K, adjoin_adjoin_left, adjoin_self _ K]
+
theorem algebra_adjoin_le_adjoin : Algebra.adjoin F S ≤ (adjoin F S).toSubalgebra :=
Algebra.adjoin_le (subset_adjoin _ _)
#align intermediate_field.algebra_adjoin_le_adjoin IntermediateField.algebra_adjoin_le_adjoin
@@ -881,6 +923,49 @@ theorem adjoin.finrank {x : L} (hx : IsIntegral K x) :
rfl
#align intermediate_field.adjoin.finrank IntermediateField.adjoin.finrank
+/-- If `K / E / F` is a field extension tower, `S ⊂ K` is such that `F(S) = K`,
+then `E(S) = K`. -/
+theorem adjoin_eq_top_of_adjoin_eq_top [Algebra E K] [IsScalarTower F E K]
+ {S : Set K} (hprim : adjoin F S = ⊤) : adjoin E S = ⊤ :=
+ restrictScalars_injective F <| by
+ rw [restrictScalars_top, ← top_le_iff, ← hprim, adjoin_le_iff,
+ coe_restrictScalars, ← adjoin_le_iff]
+
+/-- If `E / F` is a finite extension such that `E = F(α)`, then for any intermediate field `K`, the
+`F` adjoin the coefficients of `minpoly K α` is equal to `K` itself. -/
+theorem adjoin_minpoly_coeff_of_exists_primitive_element
+ [FiniteDimensional F E] (hprim : adjoin F {α} = ⊤) (K : IntermediateField F E) :
+ adjoin F ((minpoly K α).map (algebraMap K E)).frange = K := by
+ set g := (minpoly K α).map (algebraMap K E)
+ set K' : IntermediateField F E := adjoin F g.frange
+ have hsub : K' ≤ K := by
+ refine adjoin_le_iff.mpr fun x ↦ ?_
+ rw [Finset.mem_coe, mem_frange_iff]
+ rintro ⟨n, -, rfl⟩
+ rw [coeff_map]
+ apply Subtype.mem
+ have g_lifts : g ∈ lifts (algebraMap K' E) := by
+ refine g.lifts_iff_coeff_lifts.mpr fun n ↦ ?_
+ erw [Subtype.range_val]
+ by_cases hn : n ∈ g.support
+ · exact subset_adjoin F _ (mem_frange_iff.mpr ⟨n, hn, rfl⟩)
+ · exact not_mem_support_iff.mp hn ▸ zero_mem K'
+ obtain ⟨p, hp⟩ := g.lifts_and_natDegree_eq_and_monic
+ g_lifts ((minpoly.monic <| isIntegral_of_finite K α).map _)
+ have dvd_p : minpoly K' α ∣ p
+ · apply minpoly.dvd
+ rw [aeval_def, eval₂_eq_eval_map, hp.1, ← eval₂_eq_eval_map, ← aeval_def]
+ exact minpoly.aeval K α
+ have finrank_eq : ∀ K : IntermediateField F E, finrank K E = natDegree (minpoly K α)
+ · intro K
+ have := adjoin.finrank (isIntegral_of_finite K α)
+ erw [adjoin_eq_top_of_adjoin_eq_top F hprim, finrank_top K E] at this
+ exact this
+ refine eq_of_le_of_finrank_le' hsub ?_
+ simp_rw [finrank_eq]
+ convert natDegree_le_of_dvd dvd_p hp.2.2.ne_zero using 1
+ rw [hp.2.1, natDegree_map]
+
theorem _root_.minpoly.natDegree_le (x : L) [FiniteDimensional K L] :
(minpoly K x).natDegree ≤ finrank K L :=
le_of_eq_of_le (IntermediateField.adjoin.finrank (isIntegral_of_finite _ _)).symm
@@ -1236,6 +1321,20 @@ instance finiteDimensional_iSup_of_finite {ι : Type*} {t : ι → IntermediateF
exact IntermediateField.finiteDimensional_sup _ _
#align intermediate_field.finite_dimensional_supr_of_finite IntermediateField.finiteDimensional_iSup_of_finite
+/-- If `L / K` is an algebraic extension, then for any finite subset `S` of `L`, `K(S) / K` is a
+finite extension. A direct corollary of `finiteDimensional_iSup_of_finite`. -/
+theorem finiteDimensional_adjoin_of_finite_of_isAlgebraic
+ (halg : Algebra.IsAlgebraic K L) (S : Set L) [Finite S] :
+ FiniteDimensional K (adjoin K S) := by
+ let t : S → IntermediateField K L := fun x ↦ K⟮x.1⟯
+ have h : ∀ x : S, FiniteDimensional K (t x) := fun x ↦
+ adjoin.finiteDimensional <| isAlgebraic_iff_isIntegral.1 (halg x.1)
+ have hfin := finiteDimensional_iSup_of_finite (t := t)
+ have := (gc (F := K) (E := L)).l_iSup (f := fun (x : S) ↦ {x.1})
+ rw [Set.iSup_eq_iUnion, Set.iUnion_singleton_eq_range, Subtype.range_coe_subtype,
+ Set.setOf_mem_eq] at this
+ rwa [← this] at hfin
+
instance finiteDimensional_iSup_of_finset {ι : Type*} {f : ι → IntermediateField K L}
/-Porting note: changed `h` from `∀ i ∈ s, FiniteDimensional K (f i)` because this caused an
error. See `finiteDimensional_iSup_of_finset'` for a stronger version, that was the one
Diff
@@ -791,8 +791,7 @@ theorem bot_eq_top_of_finrank_adjoin_le_one [FiniteDimensional F E]
theorem subsingleton_of_finrank_adjoin_le_one [FiniteDimensional F E]
(h : ∀ x : E, finrank F F⟮x⟯ ≤ 1) : Subsingleton (IntermediateField F E) :=
subsingleton_of_bot_eq_top (bot_eq_top_of_finrank_adjoin_le_one h)
-#align intermediate_field.subsingleton_of_finrank_adjoin_le_one
- IntermediateField.subsingleton_of_finrank_adjoin_le_one
+#align intermediate_field.subsingleton_of_finrank_adjoin_le_one IntermediateField.subsingleton_of_finrank_adjoin_le_one
end AdjoinRank
refactor: golf Normal.of_isSplittingField (#8004)
Before: Construction only imports Normal, which transitively imports IsSplittingField
Now: Normal imports Construction, Construction only imports IsSplittingField
So no extra transitive import is added to any file other than Construction and Normal.
As a consequence, Polynomial.SplittingField.instNormal is moved from Construction to Normal.
adjoin_rootSet_eq_range is added to IsSplittingField.
splits_of_comp in Splits is extracted from splits_of_splits in IsSplittingField.
Source of proof: https://math.stackexchange.com/a/2585087/12932
Move Algebra.adjoin.liftSingleton from IsAlgClosed/Basic to Adjoin/Field in order to speed up lift_of_splits (renamed to add namespace Polynomial).
Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>
Diff
@@ -904,6 +904,12 @@ noncomputable def algHomAdjoinIntegralEquiv (h : IsIntegral F α) :
rw [adjoin.powerBasis_gen, minpoly_gen, Equiv.refl_apply])
#align intermediate_field.alg_hom_adjoin_integral_equiv IntermediateField.algHomAdjoinIntegralEquiv
+lemma algHomAdjoinIntegralEquiv_symm_apply_gen (h : IsIntegral F α)
+ (x : { x // x ∈ (minpoly F α).aroots K }) :
+ (algHomAdjoinIntegralEquiv F h).symm x (AdjoinSimple.gen F α) = x :=
+ (adjoin.powerBasis h).lift_gen x.val <| by
+ rw [adjoin.powerBasis_gen, minpoly_gen]; exact (mem_aroots.mp x.2).2
+
/-- Fintype of algebra homomorphism `F⟮α⟯ →ₐ[F] K` -/
noncomputable def fintypeOfAlgHomAdjoinIntegral (h : IsIntegral F α) : Fintype (F⟮α⟯ →ₐ[F] K) :=
PowerBasis.AlgHom.fintype (adjoin.powerBasis h)
refactor(Algebra/Algebra/Subalgebra/Basic): use a better defeq for ⊥ : Subalgebra R A (#8038)
And the same thing for StarSubalgebra R A. IntermediateField was already handled in #7957.
As a result, nine (obvious) lemmas are now true by definition.
This slightly adjusts the statement of Algebra.toSubmodule_bot to make it simpler and true by definition; the original statement can be recovered by rewriting by Submodule.one_eq_span, which I've had to add in some downstream proofs.
Diff
@@ -88,12 +88,11 @@ instance : CompleteLattice (IntermediateField F E) where
instance : Inhabited (IntermediateField F E) :=
⟨⊤⟩
-theorem coe_bot : ↑(⊥ : IntermediateField F E) = Set.range (algebraMap F E) :=
- Algebra.coe_bot
+theorem coe_bot : ↑(⊥ : IntermediateField F E) = Set.range (algebraMap F E) := rfl
#align intermediate_field.coe_bot IntermediateField.coe_bot
theorem mem_bot {x : E} : x ∈ (⊥ : IntermediateField F E) ↔ x ∈ Set.range (algebraMap F E) :=
- Algebra.mem_bot
+ Iff.rfl
#align intermediate_field.mem_bot IntermediateField.mem_bot
@[simp]
@@ -260,10 +259,8 @@ def topEquiv : (⊤ : IntermediateField F E) ≃ₐ[F] E :=
@[simp]
theorem restrictScalars_bot_eq_self (K : IntermediateField F E) :
- (⊥ : IntermediateField K E).restrictScalars _ = K := by
- ext x
- rw [mem_restrictScalars, mem_bot];
- exact Set.ext_iff.mp Subtype.range_coe x
+ (⊥ : IntermediateField K E).restrictScalars _ = K :=
+ SetLike.coe_injective Subtype.range_coe
#align intermediate_field.restrict_scalars_bot_eq_self IntermediateField.restrictScalars_bot_eq_self
@[simp]
refactor(FieldTheory/Adjoin): provide a better defeq for ⊥ (#7957)
This means (⊥ : IntermediateField F E).toSubalgebra = ⊥ is true by definition.
Also adds IntermediateField.map_bot now that the proof is trivial.
Diff
@@ -75,26 +75,29 @@ def gi : GaloisInsertion (adjoin F : Set E → IntermediateField F E)
choice_eq _ _ := copy_eq _ _ _
#align intermediate_field.gi IntermediateField.gi
-instance : CompleteLattice (IntermediateField F E) :=
- GaloisInsertion.liftCompleteLattice IntermediateField.gi
+instance : CompleteLattice (IntermediateField F E) where
+ __ := GaloisInsertion.liftCompleteLattice IntermediateField.gi
+ bot :=
+ { toSubalgebra := ⊥
+ inv_mem' := fun x (hx : x ∈ (⊥ : Subalgebra F E)) => show x⁻¹ ∈ (⊥ : Subalgebra F E) by
+ rw [Algebra.mem_bot] at hx ⊢
+ obtain ⟨r, rfl⟩ := hx
+ exact ⟨r⁻¹, map_inv₀ _ _⟩ }
+ bot_le x := (bot_le : ⊥ ≤ x.toSubalgebra)
instance : Inhabited (IntermediateField F E) :=
⟨⊤⟩
-theorem coe_bot : ↑(⊥ : IntermediateField F E) = Set.range (algebraMap F E) := by
- change ↑(Subfield.closure (Set.range (algebraMap F E) ∪ ∅)) = Set.range (algebraMap F E)
- rw [Set.union_empty, ← Set.image_univ, ← RingHom.map_field_closure]
- simp
+theorem coe_bot : ↑(⊥ : IntermediateField F E) = Set.range (algebraMap F E) :=
+ Algebra.coe_bot
#align intermediate_field.coe_bot IntermediateField.coe_bot
theorem mem_bot {x : E} : x ∈ (⊥ : IntermediateField F E) ↔ x ∈ Set.range (algebraMap F E) :=
- Set.ext_iff.mp coe_bot x
+ Algebra.mem_bot
#align intermediate_field.mem_bot IntermediateField.mem_bot
@[simp]
-theorem bot_toSubalgebra : (⊥ : IntermediateField F E).toSubalgebra = ⊥ := by
- ext
- rw [mem_toSubalgebra, Algebra.mem_bot, mem_bot]
+theorem bot_toSubalgebra : (⊥ : IntermediateField F E).toSubalgebra = ⊥ := rfl
#align intermediate_field.bot_to_subalgebra IntermediateField.bot_toSubalgebra
@[simp]
@@ -269,6 +272,11 @@ theorem restrictScalars_top {K : Type*} [Field K] [Algebra K E] [Algebra K F]
rfl
#align intermediate_field.restrict_scalars_top IntermediateField.restrictScalars_top
+@[simp]
+theorem map_bot {K : Type*} [Field K] [Algebra F K] (f : E →ₐ[F] K) :
+ IntermediateField.map f ⊥ = ⊥ :=
+ toSubalgebra_injective <| Algebra.map_bot _
+
theorem _root_.AlgHom.fieldRange_eq_map {K : Type*} [Field K] [Algebra F K] (f : E →ₐ[F] K) :
f.fieldRange = IntermediateField.map f ⊤ :=
SetLike.ext' Set.image_univ.symm
fix(FieldTheory): add the missing _root_ lost during porting (#7981)
You can tell this was a porting error because the #aligns were weird.
Diff
@@ -269,26 +269,26 @@ theorem restrictScalars_top {K : Type*} [Field K] [Algebra K E] [Algebra K F]
rfl
#align intermediate_field.restrict_scalars_top IntermediateField.restrictScalars_top
-theorem AlgHom.fieldRange_eq_map {K : Type*} [Field K] [Algebra F K] (f : E →ₐ[F] K) :
+theorem _root_.AlgHom.fieldRange_eq_map {K : Type*} [Field K] [Algebra F K] (f : E →ₐ[F] K) :
f.fieldRange = IntermediateField.map f ⊤ :=
SetLike.ext' Set.image_univ.symm
-#align alg_hom.field_range_eq_map IntermediateField.AlgHom.fieldRange_eq_map
+#align alg_hom.field_range_eq_map AlgHom.fieldRange_eq_map
-theorem AlgHom.map_fieldRange {K L : Type*} [Field K] [Field L] [Algebra F K] [Algebra F L]
+theorem _root_.AlgHom.map_fieldRange {K L : Type*} [Field K] [Field L] [Algebra F K] [Algebra F L]
(f : E →ₐ[F] K) (g : K →ₐ[F] L) : f.fieldRange.map g = (g.comp f).fieldRange :=
SetLike.ext' (Set.range_comp g f).symm
-#align alg_hom.map_field_range IntermediateField.AlgHom.map_fieldRange
+#align alg_hom.map_field_range AlgHom.map_fieldRange
-theorem AlgHom.fieldRange_eq_top {K : Type*} [Field K] [Algebra F K] {f : E →ₐ[F] K} :
+theorem _root_.AlgHom.fieldRange_eq_top {K : Type*} [Field K] [Algebra F K] {f : E →ₐ[F] K} :
f.fieldRange = ⊤ ↔ Function.Surjective f :=
SetLike.ext'_iff.trans Set.range_iff_surjective
-#align alg_hom.field_range_eq_top IntermediateField.AlgHom.fieldRange_eq_top
+#align alg_hom.field_range_eq_top AlgHom.fieldRange_eq_top
@[simp]
-theorem AlgEquiv.fieldRange_eq_top {K : Type*} [Field K] [Algebra F K] (f : E ≃ₐ[F] K) :
+theorem _root_.AlgEquiv.fieldRange_eq_top {K : Type*} [Field K] [Algebra F K] (f : E ≃ₐ[F] K) :
(f : E →ₐ[F] K).fieldRange = ⊤ :=
AlgHom.fieldRange_eq_top.mpr f.surjective
-#align alg_equiv.field_range_eq_top IntermediateField.AlgEquiv.fieldRange_eq_top
+#align alg_equiv.field_range_eq_top AlgEquiv.fieldRange_eq_top
end Lattice
Diff
@@ -25,7 +25,7 @@ For example, `Algebra.adjoin K {x}` might not include `x⁻¹`.
## Notation
- - `F⟮α⟯`: adjoin a single element `α` to `F`.
+ - `F⟮α⟯`: adjoin a single element `α` to `F` (in scope `IntermediateField`).
-/
set_option autoImplicit true
@@ -469,7 +469,7 @@ where
/-- If `x₁ x₂ ... xₙ : E` then `F⟮x₁,x₂,...,xₙ⟯` is the `IntermediateField F E`
generated by these elements. -/
-macro:max K:term "⟮" xs:term,* "⟯" : term => do ``(adjoin $K $(← mkInsertTerm xs.getElems))
+scoped macro:max K:term "⟮" xs:term,* "⟯" : term => do ``(adjoin $K $(← mkInsertTerm xs.getElems))
open Lean PrettyPrinter.Delaborator SubExpr in
@[delab app.IntermediateField.adjoin]
Diff
@@ -829,10 +829,12 @@ noncomputable def adjoinRootEquivAdjoin (h : IsIntegral F α) :
refine' Subfield.closure_le.mpr (Set.union_subset (fun x hx => _) _)
· obtain ⟨y, hy⟩ := hx
refine' ⟨y, _⟩
- rw [RingHom.comp_apply, AdjoinRoot.lift_of (aeval_gen_minpoly F α)]
+ -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
+ erw [RingHom.comp_apply, AdjoinRoot.lift_of (aeval_gen_minpoly F α)]
exact hy
· refine' Set.singleton_subset_iff.mpr ⟨AdjoinRoot.root (minpoly F α), _⟩
- rw [RingHom.comp_apply, AdjoinRoot.lift_root (aeval_gen_minpoly F α)]
+ -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
+ erw [RingHom.comp_apply, AdjoinRoot.lift_root (aeval_gen_minpoly F α)]
rfl)
#align intermediate_field.adjoin_root_equiv_adjoin IntermediateField.adjoinRootEquivAdjoin
@@ -860,7 +862,8 @@ noncomputable def adjoin.powerBasis {x : L} (hx : IsIntegral K x) : PowerBasis K
dim := (minpoly K x).natDegree
basis := powerBasisAux hx
basis_eq_pow i := by
- rw [powerBasisAux, Basis.map_apply, PowerBasis.basis_eq_pow, AlgEquiv.toLinearEquiv_apply,
+ -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
+ erw [powerBasisAux, Basis.map_apply, PowerBasis.basis_eq_pow, AlgEquiv.toLinearEquiv_apply,
AlgEquiv.map_pow, AdjoinRoot.powerBasis_gen, adjoinRootEquivAdjoin_apply_root]
#align intermediate_field.adjoin.power_basis IntermediateField.adjoin.powerBasis
Diff
@@ -829,12 +829,10 @@ noncomputable def adjoinRootEquivAdjoin (h : IsIntegral F α) :
refine' Subfield.closure_le.mpr (Set.union_subset (fun x hx => _) _)
· obtain ⟨y, hy⟩ := hx
refine' ⟨y, _⟩
- -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
- erw [RingHom.comp_apply, AdjoinRoot.lift_of (aeval_gen_minpoly F α)]
+ rw [RingHom.comp_apply, AdjoinRoot.lift_of (aeval_gen_minpoly F α)]
exact hy
· refine' Set.singleton_subset_iff.mpr ⟨AdjoinRoot.root (minpoly F α), _⟩
- -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
- erw [RingHom.comp_apply, AdjoinRoot.lift_root (aeval_gen_minpoly F α)]
+ rw [RingHom.comp_apply, AdjoinRoot.lift_root (aeval_gen_minpoly F α)]
rfl)
#align intermediate_field.adjoin_root_equiv_adjoin IntermediateField.adjoinRootEquivAdjoin
@@ -862,8 +860,7 @@ noncomputable def adjoin.powerBasis {x : L} (hx : IsIntegral K x) : PowerBasis K
dim := (minpoly K x).natDegree
basis := powerBasisAux hx
basis_eq_pow i := by
- -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
- erw [powerBasisAux, Basis.map_apply, PowerBasis.basis_eq_pow, AlgEquiv.toLinearEquiv_apply,
+ rw [powerBasisAux, Basis.map_apply, PowerBasis.basis_eq_pow, AlgEquiv.toLinearEquiv_apply,
AlgEquiv.map_pow, AdjoinRoot.powerBasis_gen, adjoinRootEquivAdjoin_apply_root]
#align intermediate_field.adjoin.power_basis IntermediateField.adjoin.powerBasis
Diff
@@ -829,10 +829,12 @@ noncomputable def adjoinRootEquivAdjoin (h : IsIntegral F α) :
refine' Subfield.closure_le.mpr (Set.union_subset (fun x hx => _) _)
· obtain ⟨y, hy⟩ := hx
refine' ⟨y, _⟩
- rw [RingHom.comp_apply, AdjoinRoot.lift_of (aeval_gen_minpoly F α)]
+ -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
+ erw [RingHom.comp_apply, AdjoinRoot.lift_of (aeval_gen_minpoly F α)]
exact hy
· refine' Set.singleton_subset_iff.mpr ⟨AdjoinRoot.root (minpoly F α), _⟩
- rw [RingHom.comp_apply, AdjoinRoot.lift_root (aeval_gen_minpoly F α)]
+ -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
+ erw [RingHom.comp_apply, AdjoinRoot.lift_root (aeval_gen_minpoly F α)]
rfl)
#align intermediate_field.adjoin_root_equiv_adjoin IntermediateField.adjoinRootEquivAdjoin
@@ -860,7 +862,8 @@ noncomputable def adjoin.powerBasis {x : L} (hx : IsIntegral K x) : PowerBasis K
dim := (minpoly K x).natDegree
basis := powerBasisAux hx
basis_eq_pow i := by
- rw [powerBasisAux, Basis.map_apply, PowerBasis.basis_eq_pow, AlgEquiv.toLinearEquiv_apply,
+ -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
+ erw [powerBasisAux, Basis.map_apply, PowerBasis.basis_eq_pow, AlgEquiv.toLinearEquiv_apply,
AlgEquiv.map_pow, AdjoinRoot.powerBasis_gen, adjoinRootEquivAdjoin_apply_root]
#align intermediate_field.adjoin.power_basis IntermediateField.adjoin.powerBasis
chore: replace minpoly.eq_of_algebraMap_eq by algebraMap_eq (#7228)
Also changes the repetitive names minpoly.minpoly_algHom/Equiv to minpoly.algHom/Equiv_eq
Diff
@@ -795,20 +795,14 @@ end AdjoinIntermediateFieldLattice
section AdjoinIntegralElement
-variable {F : Type*} [Field F] {E : Type*} [Field E] [Algebra F E] {α : E}
+variable (F : Type*) [Field F] {E : Type*} [Field E] [Algebra F E] {α : E}
variable {K : Type*} [Field K] [Algebra F K]
-theorem minpoly_gen {α : E} (h : IsIntegral F α) :
+theorem minpoly_gen (α : E) :
minpoly F (AdjoinSimple.gen F α) = minpoly F α := by
- rw [← AdjoinSimple.algebraMap_gen F α] at h
- have inj := (algebraMap F⟮α⟯ E).injective
- exact
- minpoly.eq_of_algebraMap_eq inj ((isIntegral_algebraMap_iff inj).mp h)
- (AdjoinSimple.algebraMap_gen _ _).symm
-#align intermediate_field.minpoly_gen IntermediateField.minpoly_gen
-
-variable (F)
+ rw [← minpoly.algebraMap_eq (algebraMap F⟮α⟯ E).injective, AdjoinSimple.algebraMap_gen]
+#align intermediate_field.minpoly_gen IntermediateField.minpoly_genₓ
theorem aeval_gen_minpoly (α : E) : aeval (AdjoinSimple.gen F α) (minpoly F α) = 0 := by
ext
@@ -899,7 +893,7 @@ noncomputable def algHomAdjoinIntegralEquiv (h : IsIntegral F α) :
(F⟮α⟯ →ₐ[F] K) ≃ { x // x ∈ (minpoly F α).aroots K } :=
(adjoin.powerBasis h).liftEquiv'.trans
((Equiv.refl _).subtypeEquiv fun x => by
- rw [adjoin.powerBasis_gen, minpoly_gen h, Equiv.refl_apply])
+ rw [adjoin.powerBasis_gen, minpoly_gen, Equiv.refl_apply])
#align intermediate_field.alg_hom_adjoin_integral_equiv IntermediateField.algHomAdjoinIntegralEquiv
/-- Fintype of algebra homomorphism `F⟮α⟯ →ₐ[F] K` -/
@@ -911,7 +905,7 @@ theorem card_algHom_adjoin_integral (h : IsIntegral F α) (h_sep : (minpoly F α
(h_splits : (minpoly F α).Splits (algebraMap F K)) :
@Fintype.card (F⟮α⟯ →ₐ[F] K) (fintypeOfAlgHomAdjoinIntegral F h) = (minpoly F α).natDegree := by
rw [AlgHom.card_of_powerBasis] <;>
- simp only [adjoin.powerBasis_dim, adjoin.powerBasis_gen, minpoly_gen h, h_sep, h_splits]
+ simp only [adjoin.powerBasis_dim, adjoin.powerBasis_gen, minpoly_gen, h_sep, h_splits]
#align intermediate_field.card_alg_hom_adjoin_integral IntermediateField.card_algHom_adjoin_integral
end AdjoinIntegralElement
@@ -1067,7 +1061,6 @@ noncomputable def Lifts.upperBoundAlgHom {c : Set (Lifts F E K)} (hc : IsChain (
simp only [Submonoid.coe_mul, Subsemiring.coe_toSubmonoid, Subalgebra.coe_toSubsemiring,
coe_toSubalgebra, Lifts.eq_of_le hzw, Lifts.eq_of_le hxw, Lifts.eq_of_le hyw, ← w.2.map_mul,
Submonoid.mk_mul_mk]
-
commutes' _ := AlgHom.commutes _ _
#align intermediate_field.lifts.upper_bound_alg_hom IntermediateField.Lifts.upperBoundAlgHom
@@ -1281,10 +1274,8 @@ open IntermediateField
/-- `pb.equivAdjoinSimple` is the equivalence between `K⟮pb.gen⟯` and `L` itself. -/
noncomputable def equivAdjoinSimple (pb : PowerBasis K L) : K⟮pb.gen⟯ ≃ₐ[K] L :=
- (adjoin.powerBasis pb.isIntegral_gen).equivOfMinpoly pb
- (minpoly.eq_of_algebraMap_eq (algebraMap K⟮pb.gen⟯ L).injective
- (adjoin.powerBasis pb.isIntegral_gen).isIntegral_gen
- (by rw [adjoin.powerBasis_gen, AdjoinSimple.algebraMap_gen]))
+ (adjoin.powerBasis pb.isIntegral_gen).equivOfMinpoly pb <| by
+ rw [adjoin.powerBasis_gen, minpoly_gen]
#align power_basis.equiv_adjoin_simple PowerBasis.equivAdjoinSimple
@[simp]
Diff
@@ -312,8 +312,8 @@ theorem adjoin.range_algebraMap_subset : Set.range (algebraMap F E) ⊆ adjoin F
exact adjoin.algebraMap_mem F S f
#align intermediate_field.adjoin.range_algebra_map_subset IntermediateField.adjoin.range_algebraMap_subset
-instance adjoin.fieldCoe : CoeTC F (adjoin F S)
- where coe x := ⟨algebraMap F E x, adjoin.algebraMap_mem F S x⟩
+instance adjoin.fieldCoe : CoeTC F (adjoin F S) where
+ coe x := ⟨algebraMap F E x, adjoin.algebraMap_mem F S x⟩
#align intermediate_field.adjoin.field_coe IntermediateField.adjoin.fieldCoe
theorem subset_adjoin : S ⊆ adjoin F S := fun _ hx => Subfield.subset_closure (Or.inr hx)
Diff
@@ -896,7 +896,7 @@ end PowerBasis
/-- Algebra homomorphism `F⟮α⟯ →ₐ[F] K` are in bijection with the set of roots
of `minpoly α` in `K`. -/
noncomputable def algHomAdjoinIntegralEquiv (h : IsIntegral F α) :
- (F⟮α⟯ →ₐ[F] K) ≃ { x // x ∈ ((minpoly F α).map (algebraMap F K)).roots } :=
+ (F⟮α⟯ →ₐ[F] K) ≃ { x // x ∈ (minpoly F α).aroots K } :=
(adjoin.powerBasis h).liftEquiv'.trans
((Equiv.refl _).subtypeEquiv fun x => by
rw [adjoin.powerBasis_gen, minpoly_gen h, Equiv.refl_apply])
@@ -1099,15 +1099,16 @@ noncomputable def Lifts.liftOfSplits (x : Lifts F E K) {s : E} (h1 : IsIntegral
splits_of_splits_of_dvd _ (map_ne_zero (minpoly.ne_zero h1))
((splits_map_iff _ _).mpr (by convert h2; exact RingHom.ext fun y => x.2.commutes y))
(minpoly.dvd_map_of_isScalarTower _ _ _)
- let alg : Algebra {y // y ∈ x.fst} {y // y ∈ restrictScalars F {z // z ∈ x.fst}⟮s⟯} :=
+ letI : Algebra {y // y ∈ x.fst} {y // y ∈ restrictScalars F {z // z ∈ x.fst}⟮s⟯} :=
{z // z ∈ x.fst}⟮s⟯.toSubalgebra.algebra
+ letI := x.2.toRingHom.toAlgebra
⟨x.1⟮s⟯.restrictScalars F,
(@algHomEquivSigma F x.1 (x.1⟮s⟯.restrictScalars F) K _ _ _ _ _ _ _
(IntermediateField.algebra x.1⟮s⟯) (IsScalarTower.of_algebraMap_eq fun _ => rfl)).invFun
⟨x.2,
- (@algHomAdjoinIntegralEquiv x.1 _ E _ _ s K _ x.2.toRingHom.toAlgebra h3).invFun
+ (@algHomAdjoinIntegralEquiv x.1 _ E _ _ s K _ _ h3).invFun
⟨rootOfSplits x.2.toRingHom key (ne_of_gt (minpoly.degree_pos h3)), by
- rw [mem_roots (map_ne_zero (minpoly.ne_zero h3)), IsRoot, ← eval₂_eq_eval_map]
+ rw [mem_aroots, and_iff_right (minpoly.ne_zero h3)]
exact map_rootOfSplits x.2.toRingHom key (ne_of_gt (minpoly.degree_pos h3))⟩⟩⟩
#align intermediate_field.lifts.lift_of_splits IntermediateField.Lifts.liftOfSplits
@@ -1128,8 +1129,8 @@ theorem Lifts.le_lifts_of_splits (x : Lifts F E K) {s : E} (h1 : IsIntegral F s)
(minpoly.dvd_map_of_isScalarTower _ _ _)
have I2 := (ne_of_gt (minpoly.degree_pos I1))
have I3 : rootOfSplits (AlgHom.toRingHom x.2) key (ne_of_gt (minpoly.degree_pos I1)) ∈
- (Polynomial.map (algebraMap x.1 K) (minpoly x.1 s)).roots := by
- rw [mem_roots (map_ne_zero (minpoly.ne_zero I1)), IsRoot, ← eval₂_eq_eval_map]
+ (minpoly x.1 s).aroots K := by
+ rw [mem_aroots, and_iff_right (minpoly.ne_zero I1)]
exact map_rootOfSplits x.2.toRingHom key (ne_of_gt (minpoly.degree_pos I1))
let φ : x.1⟮s⟯ →ₐ[x.1] K := ((algHomAdjoinIntegralEquiv x.1 I1).invFun
⟨rootOfSplits (AlgHom.toRingHom x.2) key I2, I3⟩)
Diff
@@ -1090,7 +1090,6 @@ theorem Lifts.exists_upper_bound (c : Set (Lifts F E K)) (hc : IsChain (· ≤
exact congr_arg z.2 (Subtype.ext hst)⟩
#align intermediate_field.lifts.exists_upper_bound IntermediateField.Lifts.exists_upper_bound
-set_option maxHeartbeats 350000 in
-- Porting note: instance `alg` added by hand. The proof is very slow.
/-- Extend a lift `x : Lifts F E K` to an element `s : E` whose conjugates are all in `K` -/
noncomputable def Lifts.liftOfSplits (x : Lifts F E K) {s : E} (h1 : IsIntegral F s)
@@ -1204,7 +1203,6 @@ theorem sup_toSubalgebra [h1 : FiniteDimensional K E1] [h2 : FiniteDimensional K
(Field.toIsField K)
#align intermediate_field.sup_to_subalgebra IntermediateField.sup_toSubalgebra
-set_option synthInstance.maxHeartbeats 30000 in
instance finiteDimensional_sup [h1 : FiniteDimensional K E1] [h2 : FiniteDimensional K E2] :
FiniteDimensional K (E1 ⊔ E2 : IntermediateField K L) := by
let g := Algebra.TensorProduct.productMap E1.val E2.val
Diff
@@ -45,8 +45,7 @@ variable (F : Type*) [Field F] {E : Type*} [Field E] [Algebra F E] (S : Set E)
/-- `adjoin F S` extends a field `F` by adjoining a set `S ⊆ E`. -/
def adjoin : IntermediateField F E :=
{ Subfield.closure (Set.range (algebraMap F E) ∪ S) with
- algebraMap_mem' := fun x => Subfield.subset_closure (Or.inl (Set.mem_range_self x))
- neg_mem' := fun x => by simp }
+ algebraMap_mem' := fun x => Subfield.subset_closure (Or.inl (Set.mem_range_self x)) }
#align intermediate_field.adjoin IntermediateField.adjoin
end AdjoinDef
@@ -427,9 +426,7 @@ theorem adjoin_eq_algebra_adjoin (inv_mem : ∀ x ∈ Algebra.adjoin F S, x⁻¹
(adjoin F S).toSubalgebra = Algebra.adjoin F S :=
le_antisymm
(show adjoin F S ≤
- { Algebra.adjoin F
- S with
- neg_mem' := fun _ => (Algebra.adjoin F S).neg_mem
+ { Algebra.adjoin F S with
inv_mem' := inv_mem }
from adjoin_le_iff.mpr Algebra.subset_adjoin)
(algebra_adjoin_le_adjoin _ _)
@@ -598,9 +595,6 @@ theorem adjoin_simple_isCompactElement (x : E) : IsCompactElement F⟮x⟯ := by
rintro x₁ x₂ ⟨-, ⟨F₁, rfl⟩, ⟨-, ⟨hF₁, rfl⟩, hx₁⟩⟩ ⟨-, ⟨F₂, rfl⟩, ⟨-, ⟨hF₂, rfl⟩, hx₂⟩⟩
obtain ⟨F₃, hF₃, h₁₃, h₂₃⟩ := hs F₁ hF₁ F₂ hF₂
exact mem_iUnion_of_mem F₃ (mem_iUnion_of_mem hF₃ (F₃.add_mem (h₁₃ hx₁) (h₂₃ hx₂)))
- neg_mem' := by
- rintro x ⟨-, ⟨E, rfl⟩, ⟨-, ⟨hE, rfl⟩, hx⟩⟩
- exact mem_iUnion_of_mem E (mem_iUnion_of_mem hE (E.neg_mem hx))
mul_mem' := by
rintro x₁ x₂ ⟨-, ⟨F₁, rfl⟩, ⟨-, ⟨hF₁, rfl⟩, hx₁⟩⟩ ⟨-, ⟨F₂, rfl⟩, ⟨-, ⟨hF₂, rfl⟩, hx₂⟩⟩
obtain ⟨F₃, hF₃, h₁₃, h₂₃⟩ := hs F₁ hF₁ F₂ hF₂
@@ -1040,7 +1034,6 @@ def Lifts.upperBoundIntermediateField {c : Set (Lifts F E K)} (hc : IsChain (·
carrier s := ∃ x : Lifts F E K, x ∈ insert ⊥ c ∧ (s ∈ x.1 : Prop)
zero_mem' := ⟨⊥, Set.mem_insert ⊥ c, zero_mem ⊥⟩
one_mem' := ⟨⊥, Set.mem_insert ⊥ c, one_mem ⊥⟩
- neg_mem' := by rintro _ ⟨x, y, h⟩; exact ⟨x, ⟨y, x.1.neg_mem h⟩⟩
inv_mem' := by rintro _ ⟨x, y, h⟩; exact ⟨x, ⟨y, x.1.inv_mem h⟩⟩
add_mem' := by
rintro _ _ ⟨x, hx, ha⟩ ⟨y, hy, hb⟩
fix: disable autoImplicit globally (#6528)
Autoimplicits are highly controversial and also defeat the performance-improving work in #6474.
The intent of this PR is to make autoImplicit opt-in on a per-file basis, by disabling it in the lakefile and enabling it again with set_option autoImplicit true in the few files that rely on it.
That also keeps this PR small, as opposed to attempting to "fix" files to not need it any more.
I claim that many of the uses of autoImplicit in these files are accidental; situations such as:
- Assuming
variablesare in scope, but pasting the lemma in the wrong section - Pasting in a lemma from a scratch file without checking to see if the variable names are consistent with the rest of the file
- Making a copy-paste error between lemmas and forgetting to add an explicit arguments.
Having set_option autoImplicit false as the default prevents these types of mistake being made in the 90% of files where autoImplicits are not used at all, and causes them to be caught by CI during review.
I think there were various points during the port where we encouraged porters to delete the universes u v lines; I think having autoparams for universe variables only would cover a lot of the cases we actually use them, while avoiding any real shortcomings.
A Zulip poll (after combining overlapping votes accordingly) was in favor of this change with 5:5:18 as the no:dontcare:yes vote ratio.
While this PR was being reviewed, a handful of files gained some more likely-accidental autoImplicits. In these places, set_option autoImplicit true has been placed locally within a section, rather than at the top of the file.
Diff
@@ -28,6 +28,8 @@ For example, `Algebra.adjoin K {x}` might not include `x⁻¹`.
- `F⟮α⟯`: adjoin a single element `α` to `F`.
-/
+set_option autoImplicit true
+
open FiniteDimensional Polynomial
chore: banish Type _ and Sort _ (#6499)
We remove all possible occurences of Type _ and Sort _ in favor of Type* and Sort*.
This has nice performance benefits.
Diff
@@ -37,7 +37,7 @@ namespace IntermediateField
section AdjoinDef
-variable (F : Type _) [Field F] {E : Type _} [Field E] [Algebra F E] (S : Set E)
+variable (F : Type*) [Field F] {E : Type*} [Field E] [Algebra F E] (S : Set E)
--Porting note: not adding `neg_mem'` causes an error.
/-- `adjoin F S` extends a field `F` by adjoining a set `S ⊆ E`. -/
@@ -51,7 +51,7 @@ end AdjoinDef
section Lattice
-variable {F : Type _} [Field F] {E : Type _} [Field E] [Algebra F E]
+variable {F : Type*} [Field F] {E : Type*} [Field E] [Algebra F E]
@[simp]
theorem adjoin_le_iff {S : Set E} {T : IntermediateField F E} : adjoin F S ≤ T ↔ S ≤ T :=
@@ -157,18 +157,18 @@ theorem sInf_toSubfield (S : Set (IntermediateField F E)) :
#align intermediate_field.Inf_to_subfield IntermediateField.sInf_toSubfield
@[simp, norm_cast]
-theorem coe_iInf {ι : Sort _} (S : ι → IntermediateField F E) : (↑(iInf S) : Set E) = ⋂ i, S i := by
+theorem coe_iInf {ι : Sort*} (S : ι → IntermediateField F E) : (↑(iInf S) : Set E) = ⋂ i, S i := by
simp [iInf]
#align intermediate_field.coe_infi IntermediateField.coe_iInf
@[simp]
-theorem iInf_toSubalgebra {ι : Sort _} (S : ι → IntermediateField F E) :
+theorem iInf_toSubalgebra {ι : Sort*} (S : ι → IntermediateField F E) :
(iInf S).toSubalgebra = ⨅ i, (S i).toSubalgebra :=
SetLike.coe_injective <| by simp [iInf]
#align intermediate_field.infi_to_subalgebra IntermediateField.iInf_toSubalgebra
@[simp]
-theorem iInf_toSubfield {ι : Sort _} (S : ι → IntermediateField F E) :
+theorem iInf_toSubfield {ι : Sort*} (S : ι → IntermediateField F E) :
(iInf S).toSubfield = ⨅ i, (S i).toSubfield :=
SetLike.coe_injective <| by simp [iInf]
#align intermediate_field.infi_to_subfield IntermediateField.iInf_toSubfield
@@ -263,28 +263,28 @@ theorem restrictScalars_bot_eq_self (K : IntermediateField F E) :
#align intermediate_field.restrict_scalars_bot_eq_self IntermediateField.restrictScalars_bot_eq_self
@[simp]
-theorem restrictScalars_top {K : Type _} [Field K] [Algebra K E] [Algebra K F]
+theorem restrictScalars_top {K : Type*} [Field K] [Algebra K E] [Algebra K F]
[IsScalarTower K F E] : (⊤ : IntermediateField F E).restrictScalars K = ⊤ :=
rfl
#align intermediate_field.restrict_scalars_top IntermediateField.restrictScalars_top
-theorem AlgHom.fieldRange_eq_map {K : Type _} [Field K] [Algebra F K] (f : E →ₐ[F] K) :
+theorem AlgHom.fieldRange_eq_map {K : Type*} [Field K] [Algebra F K] (f : E →ₐ[F] K) :
f.fieldRange = IntermediateField.map f ⊤ :=
SetLike.ext' Set.image_univ.symm
#align alg_hom.field_range_eq_map IntermediateField.AlgHom.fieldRange_eq_map
-theorem AlgHom.map_fieldRange {K L : Type _} [Field K] [Field L] [Algebra F K] [Algebra F L]
+theorem AlgHom.map_fieldRange {K L : Type*} [Field K] [Field L] [Algebra F K] [Algebra F L]
(f : E →ₐ[F] K) (g : K →ₐ[F] L) : f.fieldRange.map g = (g.comp f).fieldRange :=
SetLike.ext' (Set.range_comp g f).symm
#align alg_hom.map_field_range IntermediateField.AlgHom.map_fieldRange
-theorem AlgHom.fieldRange_eq_top {K : Type _} [Field K] [Algebra F K] {f : E →ₐ[F] K} :
+theorem AlgHom.fieldRange_eq_top {K : Type*} [Field K] [Algebra F K] {f : E →ₐ[F] K} :
f.fieldRange = ⊤ ↔ Function.Surjective f :=
SetLike.ext'_iff.trans Set.range_iff_surjective
#align alg_hom.field_range_eq_top IntermediateField.AlgHom.fieldRange_eq_top
@[simp]
-theorem AlgEquiv.fieldRange_eq_top {K : Type _} [Field K] [Algebra F K] (f : E ≃ₐ[F] K) :
+theorem AlgEquiv.fieldRange_eq_top {K : Type*} [Field K] [Algebra F K] (f : E ≃ₐ[F] K) :
(f : E →ₐ[F] K).fieldRange = ⊤ :=
AlgHom.fieldRange_eq_top.mpr f.surjective
#align alg_equiv.field_range_eq_top IntermediateField.AlgEquiv.fieldRange_eq_top
@@ -293,7 +293,7 @@ end Lattice
section AdjoinDef
-variable (F : Type _) [Field F] {E : Type _} [Field E] [Algebra F E] (S : Set E)
+variable (F : Type*) [Field F] {E : Type*} [Field E] [Algebra F E] (S : Set E)
theorem adjoin_eq_range_algebraMap_adjoin :
(adjoin F S : Set E) = Set.range (algebraMap (adjoin F S) E) :=
@@ -339,12 +339,12 @@ theorem subset_adjoin_of_subset_right {T : Set E} (H : T ⊆ S) : T ⊆ adjoin F
#align intermediate_field.subset_adjoin_of_subset_right IntermediateField.subset_adjoin_of_subset_right
@[simp]
-theorem adjoin_empty (F E : Type _) [Field F] [Field E] [Algebra F E] : adjoin F (∅ : Set E) = ⊥ :=
+theorem adjoin_empty (F E : Type*) [Field F] [Field E] [Algebra F E] : adjoin F (∅ : Set E) = ⊥ :=
eq_bot_iff.mpr (adjoin_le_iff.mpr (Set.empty_subset _))
#align intermediate_field.adjoin_empty IntermediateField.adjoin_empty
@[simp]
-theorem adjoin_univ (F E : Type _) [Field F] [Field E] [Algebra F E] :
+theorem adjoin_univ (F E : Type*) [Field F] [Field E] [Algebra F E] :
adjoin F (Set.univ : Set E) = ⊤ :=
eq_top_iff.mpr <| subset_adjoin _ _
#align intermediate_field.adjoin_univ IntermediateField.adjoin_univ
@@ -357,7 +357,7 @@ theorem adjoin_le_subfield {K : Subfield E} (HF : Set.range (algebraMap F E) ⊆
exact ⟨HF, HS⟩
#align intermediate_field.adjoin_le_subfield IntermediateField.adjoin_le_subfield
-theorem adjoin_subset_adjoin_iff {F' : Type _} [Field F'] [Algebra F' E] {S S' : Set E} :
+theorem adjoin_subset_adjoin_iff {F' : Type*} [Field F'] [Algebra F' E] {S S' : Set E} :
(adjoin F S : Set E) ⊆ adjoin F' S' ↔
Set.range (algebraMap F E) ⊆ adjoin F' S' ∧ S ⊆ adjoin F' S' :=
⟨fun h => ⟨_root_.trans (adjoin.range_algebraMap_subset _ _) h, _root_.trans
@@ -406,7 +406,7 @@ theorem adjoin_adjoin_comm (T : Set E) :
rw [adjoin_adjoin_left, adjoin_adjoin_left, Set.union_comm]
#align intermediate_field.adjoin_adjoin_comm IntermediateField.adjoin_adjoin_comm
-theorem adjoin_map {E' : Type _} [Field E'] [Algebra F E'] (f : E →ₐ[F] E') :
+theorem adjoin_map {E' : Type*} [Field E'] [Algebra F E'] (f : E →ₐ[F] E') :
(adjoin F S).map f = adjoin F (f '' S) := by
ext x
show
@@ -563,7 +563,7 @@ theorem adjoin_rootSet_isSplittingField {p : F[X]} (hp : p.Splits (algebraMap F
open scoped BigOperators
/-- A compositum of splitting fields is a splitting field -/
-theorem isSplittingField_iSup {ι : Type _} {t : ι → IntermediateField F E} {p : ι → F[X]}
+theorem isSplittingField_iSup {ι : Type*} {t : ι → IntermediateField F E} {p : ι → F[X]}
{s : Finset ι} (h0 : ∏ i in s, p i ≠ 0) (h : ∀ i ∈ s, (p i).IsSplittingField F (t i)) :
(∏ i in s, p i).IsSplittingField F (⨆ i ∈ s, t i : IntermediateField F E) := by
let K : IntermediateField F E := ⨆ i ∈ s, t i
@@ -645,7 +645,7 @@ instance : IsCompactlyGenerated (IntermediateField F E) :=
(le_antisymm (iSup_le fun i => iSup_le fun hi => adjoin_simple_le_iff.mpr hi) fun x hx =>
adjoin_simple_le_iff.mp (le_iSup_of_le x (le_iSup_of_le hx le_rfl)))⟩⟩⟩
-theorem exists_finset_of_mem_iSup {ι : Type _} {f : ι → IntermediateField F E} {x : E}
+theorem exists_finset_of_mem_iSup {ι : Type*} {f : ι → IntermediateField F E} {x : E}
(hx : x ∈ ⨆ i, f i) : ∃ s : Finset ι, x ∈ ⨆ i ∈ s, f i := by
have := (adjoin_simple_isCompactElement x : IsCompactElement F⟮x⟯).exists_finset_of_le_iSup
(IntermediateField F E) f
@@ -653,7 +653,7 @@ theorem exists_finset_of_mem_iSup {ι : Type _} {f : ι → IntermediateField F
exact this hx
#align intermediate_field.exists_finset_of_mem_supr IntermediateField.exists_finset_of_mem_iSup
-theorem exists_finset_of_mem_supr' {ι : Type _} {f : ι → IntermediateField F E} {x : E}
+theorem exists_finset_of_mem_supr' {ι : Type*} {f : ι → IntermediateField F E} {x : E}
(hx : x ∈ ⨆ i, f i) : ∃ s : Finset (Σ i, f i), x ∈ ⨆ i ∈ s, F⟮(i.2 : E)⟯ := by
-- Porting note: writing `fun i x h => ...` does not work.
refine' exists_finset_of_mem_iSup (SetLike.le_def.mp (iSup_le (fun i => _)) hx)
@@ -661,7 +661,7 @@ theorem exists_finset_of_mem_supr' {ι : Type _} {f : ι → IntermediateField F
exact SetLike.le_def.mp (le_iSup_of_le ⟨i, x, h⟩ (by simp)) (mem_adjoin_simple_self F x)
#align intermediate_field.exists_finset_of_mem_supr' IntermediateField.exists_finset_of_mem_supr'
-theorem exists_finset_of_mem_supr'' {ι : Type _} {f : ι → IntermediateField F E}
+theorem exists_finset_of_mem_supr'' {ι : Type*} {f : ι → IntermediateField F E}
(h : ∀ i, Algebra.IsAlgebraic F (f i)) {x : E} (hx : x ∈ ⨆ i, f i) :
∃ s : Finset (Σ i, f i), x ∈ ⨆ i ∈ s, adjoin F ((minpoly F (i.2 : _)).rootSet E) := by
-- Porting note: writing `fun i x1 hx1 => ...` does not work.
@@ -680,7 +680,7 @@ end AdjoinDef
section AdjoinIntermediateFieldLattice
-variable {F : Type _} [Field F] {E : Type _} [Field E] [Algebra F E] {α : E} {S : Set E}
+variable {F : Type*} [Field F] {E : Type*} [Field E] [Algebra F E] {α : E} {S : Set E}
@[simp]
theorem adjoin_eq_bot_iff : adjoin F S = ⊥ ↔ S ⊆ (⊥ : IntermediateField F E) := by
@@ -799,9 +799,9 @@ end AdjoinIntermediateFieldLattice
section AdjoinIntegralElement
-variable {F : Type _} [Field F] {E : Type _} [Field E] [Algebra F E] {α : E}
+variable {F : Type*} [Field F] {E : Type*} [Field E] [Algebra F E] {α : E}
-variable {K : Type _} [Field K] [Algebra F K]
+variable {K : Type*} [Field K] [Algebra F K]
theorem minpoly_gen {α : E} (h : IsIntegral F α) :
minpoly F (AdjoinSimple.gen F α) = minpoly F α := by
@@ -853,7 +853,7 @@ theorem adjoinRootEquivAdjoin_apply_root (h : IsIntegral F α) :
section PowerBasis
-variable {L : Type _} [Field L] [Algebra K L]
+variable {L : Type*} [Field L] [Algebra K L]
/-- The elements `1, x, ..., x ^ (d - 1)` form a basis for `K⟮x⟯`,
where `d` is the degree of the minimal polynomial of `x`. -/
@@ -922,7 +922,7 @@ end AdjoinIntegralElement
section Induction
-variable {F : Type _} [Field F] {E : Type _} [Field E] [Algebra F E]
+variable {F : Type*} [Field F] {E : Type*} [Field E] [Algebra F E]
/-- An intermediate field `S` is finitely generated if there exists `t : Finset E` such that
`IntermediateField.adjoin F t = S`. -/
@@ -979,7 +979,7 @@ end Induction
section AlgHomMkAdjoinSplits
-variable (F E K : Type _) [Field F] [Field E] [Field K] [Algebra F E] [Algebra F K] {S : Set E}
+variable (F E K : Type*) [Field F] [Field E] [Field K] [Algebra F E] [Algebra F K] {S : Set E}
/-- Lifts `L → K` of `F → K` -/
def Lifts :=
@@ -1181,7 +1181,7 @@ end AlgHomMkAdjoinSplits
section Supremum
-variable {K L : Type _} [Field K] [Field L] [Algebra K L] (E1 E2 : IntermediateField K L)
+variable {K L : Type*} [Field K] [Field L] [Algebra K L] (E1 E2 : IntermediateField K L)
theorem le_sup_toSubalgebra : E1.toSubalgebra ⊔ E2.toSubalgebra ≤ (E1 ⊔ E2).toSubalgebra :=
sup_le (show E1 ≤ E1 ⊔ E2 from le_sup_left) (show E2 ≤ E1 ⊔ E2 from le_sup_right)
@@ -1220,7 +1220,7 @@ instance finiteDimensional_sup [h1 : FiniteDimensional K E1] [h2 : FiniteDimensi
rw [Algebra.TensorProduct.productMap_range, E1.range_val, E2.range_val, sup_toSubalgebra]
#align intermediate_field.finite_dimensional_sup IntermediateField.finiteDimensional_sup
-instance finiteDimensional_iSup_of_finite {ι : Type _} {t : ι → IntermediateField K L}
+instance finiteDimensional_iSup_of_finite {ι : Type*} {t : ι → IntermediateField K L}
[h : Finite ι] [∀ i, FiniteDimensional K (t i)] :
FiniteDimensional K (⨆ i, t i : IntermediateField K L) := by
rw [← iSup_univ]
@@ -1236,7 +1236,7 @@ instance finiteDimensional_iSup_of_finite {ι : Type _} {t : ι → Intermediate
exact IntermediateField.finiteDimensional_sup _ _
#align intermediate_field.finite_dimensional_supr_of_finite IntermediateField.finiteDimensional_iSup_of_finite
-instance finiteDimensional_iSup_of_finset {ι : Type _} {f : ι → IntermediateField K L}
+instance finiteDimensional_iSup_of_finset {ι : Type*} {f : ι → IntermediateField K L}
/-Porting note: changed `h` from `∀ i ∈ s, FiniteDimensional K (f i)` because this caused an
error. See `finiteDimensional_iSup_of_finset'` for a stronger version, that was the one
used in mathlib3.-/
@@ -1249,7 +1249,7 @@ instance finiteDimensional_iSup_of_finset {ι : Type _} {f : ι → Intermediate
exact this.symm ▸ IntermediateField.finiteDimensional_iSup_of_finite
#align intermediate_field.finite_dimensional_supr_of_finset IntermediateField.finiteDimensional_iSup_of_finset
-theorem finiteDimensional_iSup_of_finset' {ι : Type _} {f : ι → IntermediateField K L}
+theorem finiteDimensional_iSup_of_finset' {ι : Type*} {f : ι → IntermediateField K L}
/-Porting note: this was the mathlib3 version. Using `[h : ...]`, as in mathlib3, causes the
error "invalid parametric local instance".-/
{s : Finset ι} (h : ∀ i, i ∈ s → FiniteDimensional K (f i)) :
@@ -1261,7 +1261,7 @@ theorem finiteDimensional_iSup_of_finset' {ι : Type _} {f : ι → Intermediate
exact this.symm ▸ IntermediateField.finiteDimensional_iSup_of_finite
/-- A compositum of algebraic extensions is algebraic -/
-theorem isAlgebraic_iSup {ι : Type _} {f : ι → IntermediateField K L}
+theorem isAlgebraic_iSup {ι : Type*} {f : ι → IntermediateField K L}
(h : ∀ i, Algebra.IsAlgebraic K (f i)) :
Algebra.IsAlgebraic K (⨆ i, f i : IntermediateField K L) := by
rintro ⟨x, hx⟩
@@ -1279,7 +1279,7 @@ end IntermediateField
section PowerBasis
-variable {K L : Type _} [Field K] [Field L] [Algebra K L]
+variable {K L : Type*} [Field K] [Field L] [Algebra K L]
namespace PowerBasis
fix: fill in "in" in maxHeartbeats (#6206)
git ls-files '*.lean' | xargs sed -i "s=maxHeartbeats [0-9]*$=& in="
Affected files:
Mathlib/CategoryTheory/Monoidal/Braided.lean:479:set_option maxHeartbeats 400000
Mathlib/FieldTheory/Adjoin.lean:1212:set_option synthInstance.maxHeartbeats 30000
Mathlib/FieldTheory/IsAlgClosed/Basic.lean:302:set_option maxHeartbeats 800000
Mathlib/FieldTheory/IsAlgClosed/Basic.lean:303:set_option synthInstance.maxHeartbeats 400000
Mathlib/RepresentationTheory/GroupCohomology/Basic.lean:202:set_option maxHeartbeats 6400000
Mathlib/RepresentationTheory/GroupCohomology/Resolution.lean:345:set_option maxHeartbeats 800000
Mathlib/Topology/MetricSpace/GromovHausdorff.lean:415:set_option maxHeartbeats 300000
Diff
@@ -1209,7 +1209,7 @@ theorem sup_toSubalgebra [h1 : FiniteDimensional K E1] [h2 : FiniteDimensional K
(Field.toIsField K)
#align intermediate_field.sup_to_subalgebra IntermediateField.sup_toSubalgebra
-set_option synthInstance.maxHeartbeats 30000
+set_option synthInstance.maxHeartbeats 30000 in
instance finiteDimensional_sup [h1 : FiniteDimensional K E1] [h2 : FiniteDimensional K E2] :
FiniteDimensional K (E1 ⊔ E2 : IntermediateField K L) := by
let g := Algebra.TensorProduct.productMap E1.val E2.val
chore(FieldTheory/Adjoin): remove unnecessary assumptions in minpolynatDegree_le and minpoly.degree_le (#6152)
Also
- fix the names of
minpoly.natDegree_leandminpoly.degree_le - rename
minpoly.ne_zero_of_finite_field_extensiontominpoly.ne_zero_of_finite - reduce typeclass assumptions of some lemmas in
RingTheory/Algebraic - add two lemmas
isIntegral_of_finiteandisAlgebraic_of_finite - move
Algebra.isIntegral_of_finitetoRingTheory/IntegralClosure
Diff
@@ -884,15 +884,16 @@ theorem adjoin.finrank {x : L} (hx : IsIntegral K x) :
rfl
#align intermediate_field.adjoin.finrank IntermediateField.adjoin.finrank
-theorem minpoly.natDegree_le {x : L} [FiniteDimensional K L] (hx : IsIntegral K x) :
+theorem _root_.minpoly.natDegree_le (x : L) [FiniteDimensional K L] :
(minpoly K x).natDegree ≤ finrank K L :=
- le_of_eq_of_le (IntermediateField.adjoin.finrank hx).symm K⟮x⟯.toSubmodule.finrank_le
-#align minpoly.nat_degree_le IntermediateField.minpoly.natDegree_le
+ le_of_eq_of_le (IntermediateField.adjoin.finrank (isIntegral_of_finite _ _)).symm
+ K⟮x⟯.toSubmodule.finrank_le
+#align minpoly.nat_degree_le minpoly.natDegree_le
-theorem minpoly.degree_le {x : L} [FiniteDimensional K L] (hx : IsIntegral K x) :
+theorem _root_.minpoly.degree_le (x : L) [FiniteDimensional K L] :
(minpoly K x).degree ≤ finrank K L :=
- degree_le_of_natDegree_le (minpoly.natDegree_le hx)
-#align minpoly.degree_le IntermediateField.minpoly.degree_le
+ degree_le_of_natDegree_le (minpoly.natDegree_le x)
+#align minpoly.degree_le minpoly.degree_le
end PowerBasis
Diff
@@ -2,17 +2,14 @@
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-
-! This file was ported from Lean 3 source module field_theory.adjoin
-! leanprover-community/mathlib commit df76f43357840485b9d04ed5dee5ab115d420e87
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathlib.FieldTheory.IntermediateField
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.SplittingField.IsSplittingField
import Mathlib.RingTheory.TensorProduct
+#align_import field_theory.adjoin from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
+
/-!
# Adjoining Elements to Fields
Diff
@@ -629,7 +629,7 @@ theorem adjoin_finset_isCompactElement (S : Finset E) :
-- Porting note: `exact` or `apply` timeout here
refine' le_antisymm (adjoin_le_iff.mpr fun x hx => SetLike.mem_coe.mpr
(adjoin_simple_le_iff.mp (le_iSup_of_le x (le_iSup_iff.2 (fun E1 hE1 => hE1 hx)))))
- (iSup_le fun x => iSup_le fun hx => adjoin_simple_le_iff.mpr (subset_adjoin F S hx))
+ (iSup_le fun x => iSup_le fun hx => adjoin_simple_le_iff.mpr (subset_adjoin F S hx))
rw [key, ← Finset.sup_eq_iSup]
exact finset_sup_compact_of_compact S fun x _ => adjoin_simple_isCompactElement x
#align intermediate_field.adjoin_finset_is_compact_element IntermediateField.adjoin_finset_isCompactElement
chore: bump to nightly-2023-07-01 (#5409)
- depends on: https://github.com/leanprover/std4/pull/163
- depends on: #5584
- depends on: #5715
- depends on: #5729
- depends on: https://github.com/leanprover/std4/pull/173
Co-authored-by: Komyyy <pol_tta@outlook.jp> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au> Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com> Co-authored-by: Mario Carneiro <di.gama@gmail.com>
Diff
@@ -1158,7 +1158,8 @@ theorem Lifts.exists_lift_of_splits (x : Lifts F E K) {s : E} (h1 : IsIntegral F
theorem algHom_mk_adjoin_splits
(hK : ∀ s ∈ S, IsIntegral F (s : E) ∧ (minpoly F s).Splits (algebraMap F K)) :
Nonempty (adjoin F S →ₐ[F] K) := by
- obtain ⟨x : Lifts F E K, hx⟩ := zorn_partialOrder Lifts.exists_upper_bound
+ obtain ⟨x, hx⟩ : ∃ m : Lifts F E K, ∀ a, m ≤ a → a = m :=
+ zorn_partialOrder Lifts.exists_upper_bound
refine'
⟨{ toFun := (fun s => x.2 ⟨s, adjoin_le_iff.mpr (fun s hs => _) s.mem⟩)
map_one' := x.2.map_one
fix: ∑' precedence (#5615)
- Also remove most superfluous parentheses around big operators (
∑,∏and variants). - roughly the used regex:
([^a-zA-Zα-ωΑ-Ω'𝓝ℳ₀𝕂ₛ)]) \(([∑∏][^()∑∏]*,[^()∑∏:]*)\) ([⊂⊆=<≤])replaced by$1 $2 $3
Diff
@@ -567,7 +567,7 @@ open scoped BigOperators
/-- A compositum of splitting fields is a splitting field -/
theorem isSplittingField_iSup {ι : Type _} {t : ι → IntermediateField F E} {p : ι → F[X]}
- {s : Finset ι} (h0 : (∏ i in s, p i) ≠ 0) (h : ∀ i ∈ s, (p i).IsSplittingField F (t i)) :
+ {s : Finset ι} (h0 : ∏ i in s, p i ≠ 0) (h : ∀ i ∈ s, (p i).IsSplittingField F (t i)) :
(∏ i in s, p i).IsSplittingField F (⨆ i ∈ s, t i : IntermediateField F E) := by
let K : IntermediateField F E := ⨆ i ∈ s, t i
have hK : ∀ i ∈ s, t i ≤ K := fun i hi => le_iSup_of_le i (le_iSup (fun _ => t i) hi)
Diff
@@ -1244,7 +1244,7 @@ instance finiteDimensional_iSup_of_finset {ι : Type _} {f : ι → Intermediate
{s : Finset ι} [h : ∀ i, FiniteDimensional K (f i)] :
FiniteDimensional K (⨆ i ∈ s, f i : IntermediateField K L) := by
haveI : ∀ i : { i // i ∈ s }, FiniteDimensional K (f i) := fun i => h i
- have : (⨆ i ∈ s, f i) = ⨆ i : { i // i ∈ s }, f i :=
+ have : ⨆ i ∈ s, f i = ⨆ i : { i // i ∈ s }, f i :=
le_antisymm (iSup_le fun i => iSup_le fun h => le_iSup (fun i : { i // i ∈ s } => f i) ⟨i, h⟩)
(iSup_le fun i => le_iSup_of_le i (le_iSup_of_le i.2 le_rfl))
exact this.symm ▸ IntermediateField.finiteDimensional_iSup_of_finite
@@ -1256,7 +1256,7 @@ theorem finiteDimensional_iSup_of_finset' {ι : Type _} {f : ι → Intermediate
{s : Finset ι} (h : ∀ i, i ∈ s → FiniteDimensional K (f i)) :
FiniteDimensional K (⨆ i ∈ s, f i : IntermediateField K L) := by
haveI : ∀ i : { i // i ∈ s }, FiniteDimensional K (f i) := fun i => h i i.2
- have : (⨆ i ∈ s, f i) = ⨆ i : { i // i ∈ s }, f i :=
+ have : ⨆ i ∈ s, f i = ⨆ i : { i // i ∈ s }, f i :=
le_antisymm (iSup_le fun i => iSup_le fun h => le_iSup (fun i : { i // i ∈ s } => f i) ⟨i, h⟩)
(iSup_le fun i => le_iSup_of_le i (le_iSup_of_le i.2 le_rfl))
exact this.symm ▸ IntermediateField.finiteDimensional_iSup_of_finite
feat(Data.Set.Basic/Data.Finset.Basic): rename insert_subset (#5450)
Currently, (for both Set and Finset) insert_subset is an iff lemma stating that insert a s ⊆ t if and only if a ∈ t and s ⊆ t. For both types, this PR renames this lemma to insert_subset_iff, and adds an insert_subset lemma that gives the implication just in the reverse direction : namely theorem insert_subset (ha : a ∈ t) (hs : s ⊆ t) : insert a s ⊆ t .
This both aligns the naming with union_subset and union_subset_iff, and removes the need for the awkward insert_subset.mpr ⟨_,_⟩ idiom. It touches a lot of files (too many to list), but in a trivial way.
Diff
@@ -397,7 +397,7 @@ theorem adjoin_insert_adjoin (x : E) :
adjoin F (insert x (adjoin F S : Set E)) = adjoin F (insert x S) :=
le_antisymm
(adjoin_le_iff.mpr
- (Set.insert_subset.mpr
+ (Set.insert_subset_iff.mpr
⟨subset_adjoin _ _ (Set.mem_insert _ _),
adjoin_le_iff.mpr (subset_adjoin_of_subset_right _ _ (Set.subset_insert _ _))⟩))
(adjoin.mono _ _ _ (Set.insert_subset_insert (subset_adjoin _ _)))
chore: change Field.toEuclideanDomain (#5266)
Modifying the definition of Field.toEuclideanDomain makes some declaration faster.
Co-authored-by: Sébastien Gouëzel
Diff
@@ -865,7 +865,6 @@ noncomputable def powerBasisAux {x : L} (hx : IsIntegral K x) :
(AdjoinRoot.powerBasis (minpoly.ne_zero hx)).basis.map (adjoinRootEquivAdjoin K hx).toLinearEquiv
#align intermediate_field.power_basis_aux IntermediateField.powerBasisAux
-set_option maxHeartbeats 300000 in
/-- The power basis `1, x, ..., x ^ (d - 1)` for `K⟮x⟯`,
where `d` is the degree of the minimal polynomial of `x`. -/
@[simps]
@@ -1098,7 +1097,7 @@ theorem Lifts.exists_upper_bound (c : Set (Lifts F E K)) (hc : IsChain (· ≤
exact congr_arg z.2 (Subtype.ext hst)⟩
#align intermediate_field.lifts.exists_upper_bound IntermediateField.Lifts.exists_upper_bound
-set_option maxHeartbeats 410000 in
+set_option maxHeartbeats 350000 in
-- Porting note: instance `alg` added by hand. The proof is very slow.
/-- Extend a lift `x : Lifts F E K` to an element `s : E` whose conjugates are all in `K` -/
noncomputable def Lifts.liftOfSplits (x : Lifts F E K) {s : E} (h1 : IsIntegral F s)
fix: improve IntermediateField.adjoin notation (#5110)
This replaces the hacky typeclass-based system I wrote before, and now there are true set notations in the elaborated terms.
Diff
@@ -455,28 +455,48 @@ theorem adjoin_induction {s : Set E} {p : E → Prop} {x} (h : x ∈ adjoin F s)
((algebraMap F E).map_one ▸ Hmap 1) Hadd Hneg Hinv Hmul
#align intermediate_field.adjoin_induction IntermediateField.adjoin_induction
---this definition of notation is courtesy of Kyle Miller on zulip
-/-- Variation on `set.insert` to enable good notation for adjoining elements to fields.
-Used to preferentially use `singleton` rather than `insert` when adjoining one element.
--/
-class Insert {α : Type _} (s : Set α) where
- /-- Variation on `set.insert`. -/
- insert : α → Set α
-#align intermediate_field.insert IntermediateField.Insert
-
-instance (priority := 1000) insertEmpty {α : Type _} : Insert (∅ : Set α)
- where insert x := @singleton _ _ inferInstance x
-#align intermediate_field.insert_empty IntermediateField.insertEmpty
-
-instance (priority := 900) insertNonempty {α : Type _} (s : Set α) : Insert s
- where insert x := _root_.Insert.insert x s
-#align intermediate_field.insert_nonempty IntermediateField.insertNonempty
-
-/-- If `S : Set E` then `F⟮S⟯` is the `IntermediateField F E` generated by `S`. -/
-notation3:max K"⟮"(l ", "* => foldr (h t => Insert.insert t h) ∅)"⟯" => adjoin K l
+/- Porting note (kmill): this notation is replacing the typeclass-based one I had previously
+written, and it gives true `{x₁, x₂, ..., xₙ}` sets in the `adjoin` term. -/
+
+open Lean in
+/-- Supporting function for the `F⟮x₁,x₂,...,xₙ⟯` adjunction notation. -/
+private partial def mkInsertTerm [Monad m] [MonadQuotation m] (xs : TSyntaxArray `term) : m Term :=
+ run 0
+where
+ run (i : Nat) : m Term := do
+ if i + 1 == xs.size then
+ ``(singleton $(xs[i]!))
+ else if i < xs.size then
+ ``(insert $(xs[i]!) $(← run (i + 1)))
+ else
+ ``(EmptyCollection.emptyCollection)
+
+/-- If `x₁ x₂ ... xₙ : E` then `F⟮x₁,x₂,...,xₙ⟯` is the `IntermediateField F E`
+generated by these elements. -/
+macro:max K:term "⟮" xs:term,* "⟯" : term => do ``(adjoin $K $(← mkInsertTerm xs.getElems))
+
+open Lean PrettyPrinter.Delaborator SubExpr in
+@[delab app.IntermediateField.adjoin]
+partial def delabAdjoinNotation : Delab := whenPPOption getPPNotation do
+ let e ← getExpr
+ guard <| e.isAppOfArity ``adjoin 6
+ let F ← withNaryArg 0 delab
+ let xs ← withNaryArg 5 delabInsertArray
+ `($F⟮$(xs.toArray),*⟯)
+where
+ delabInsertArray : DelabM (List Term) := do
+ let e ← getExpr
+ if e.isAppOfArity ``EmptyCollection.emptyCollection 2 then
+ return []
+ else if e.isAppOfArity ``singleton 4 then
+ let x ← withNaryArg 3 delab
+ return [x]
+ else if e.isAppOfArity ``insert 5 then
+ let x ← withNaryArg 3 delab
+ let xs ← withNaryArg 4 delabInsertArray
+ return x :: xs
+ else failure
--- Porting note: without the following line the linter complains.
-attribute [nolint docBlame] IntermediateField.«term_⟮_,⟯».delab
section AdjoinSimple
variable (α : E)
@@ -563,16 +583,9 @@ theorem isSplittingField_iSup {ι : Type _} {t : ι → IntermediateField F E} {
open Set CompleteLattice
-/- Porting note: added because `adjoin_simple_le_iff` below cannot be tagged `simp`.-/
-@[simp]
-theorem insert_le_iff {K : IntermediateField F E} : Insert.insert ∅ α ⊆ ↑K ↔ α ∈ K := by
- simp only [← adjoin_le_iff, ← le_eq_subset]
- exact adjoin_le_iff.trans singleton_subset_iff
-
-/- Porting note: this was tagged `simp`, but the LHS can be simplified, see `insert_le_iff` above
-for the corresponding declaration. -/
-theorem adjoin_simple_le_iff {K : IntermediateField F E} : F⟮α⟯ ≤ K ↔ α ∈ K := by
- simp
+/- Porting note: this was tagged `simp`, but the LHS can be simplified now that the notation
+has been improved. -/
+theorem adjoin_simple_le_iff {K : IntermediateField F E} : F⟮α⟯ ≤ K ↔ α ∈ K := by simp
#align intermediate_field.adjoin_simple_le_iff IntermediateField.adjoin_simple_le_iff
/-- Adjoining a single element is compact in the lattice of intermediate fields. -/