number_theory.number_field.norm ⟷ Mathlib.NumberTheory.NumberField.Norm

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -68,13 +68,36 @@ theorem norm_algebraMap [IsSeparable K L] (x : 𝓞 K) :
 #print RingOfIntegers.isUnit_norm_of_isGalois /-
 theorem isUnit_norm_of_isGalois [IsGalois K L] {x : 𝓞 L} : IsUnit (norm K x) ↔ IsUnit x := by
   classical
+  refine' ⟨fun hx => _, IsUnit.map _⟩
+  replace hx : IsUnit (algebraMap (𝓞 K) (𝓞 L) <| norm K x) := hx.map (algebraMap (𝓞 K) <| 𝓞 L)
+  refine'
+    @isUnit_of_mul_isUnit_right (𝓞 L) _
+      ⟨(univ \ {AlgEquiv.refl}).Prod fun σ : L ≃ₐ[K] L => σ x,
+        prod_mem fun σ hσ => IsIntegral.map (σ : L →+* L).toIntAlgHom x.2⟩
+      _ _
+  convert hx using 1
+  ext
+  push_cast
+  convert_to
+    ((univ \ {AlgEquiv.refl}).Prod fun σ : L ≃ₐ[K] L => σ x) *
+        ∏ σ : L ≃ₐ[K] L in {AlgEquiv.refl}, σ (x : L) =
+      _
+  · rw [prod_singleton, AlgEquiv.coe_refl, id]
+  · rw [prod_sdiff <| subset_univ _, ← norm_eq_prod_automorphisms, coe_algebra_map_norm]
 #align ring_of_integers.is_unit_norm_of_is_galois RingOfIntegers.isUnit_norm_of_isGalois
 -/
 
 #print RingOfIntegers.dvd_norm /-
 /-- If `L/K` is a finite Galois extension of fields, then, for all `(x : 𝓞 L)` we have that
 `x ∣ algebra_map (𝓞 K) (𝓞 L) (norm K x)`. -/
-theorem dvd_norm [IsGalois K L] (x : 𝓞 L) : x ∣ algebraMap (𝓞 K) (𝓞 L) (norm K x) := by classical
+theorem dvd_norm [IsGalois K L] (x : 𝓞 L) : x ∣ algebraMap (𝓞 K) (𝓞 L) (norm K x) := by
+  classical
+  have hint : ∏ σ : L ≃ₐ[K] L in univ.erase AlgEquiv.refl, σ x ∈ 𝓞 L :=
+    Subalgebra.prod_mem _ fun σ hσ =>
+      (mem_ring_of_integers _ _).2 (IsIntegral.map σ (ring_of_integers.is_integral_coe x))
+  refine' ⟨⟨_, hint⟩, Subtype.ext _⟩
+  rw [coe_algebra_map_norm K x, norm_eq_prod_automorphisms]
+  simp [← Finset.mul_prod_erase _ _ (mem_univ AlgEquiv.refl)]
 #align ring_of_integers.dvd_norm RingOfIntegers.dvd_norm
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -68,36 +68,13 @@ theorem norm_algebraMap [IsSeparable K L] (x : 𝓞 K) :
 #print RingOfIntegers.isUnit_norm_of_isGalois /-
 theorem isUnit_norm_of_isGalois [IsGalois K L] {x : 𝓞 L} : IsUnit (norm K x) ↔ IsUnit x := by
   classical
-  refine' ⟨fun hx => _, IsUnit.map _⟩
-  replace hx : IsUnit (algebraMap (𝓞 K) (𝓞 L) <| norm K x) := hx.map (algebraMap (𝓞 K) <| 𝓞 L)
-  refine'
-    @isUnit_of_mul_isUnit_right (𝓞 L) _
-      ⟨(univ \ {AlgEquiv.refl}).Prod fun σ : L ≃ₐ[K] L => σ x,
-        prod_mem fun σ hσ => IsIntegral.map (σ : L →+* L).toIntAlgHom x.2⟩
-      _ _
-  convert hx using 1
-  ext
-  push_cast
-  convert_to
-    ((univ \ {AlgEquiv.refl}).Prod fun σ : L ≃ₐ[K] L => σ x) *
-        ∏ σ : L ≃ₐ[K] L in {AlgEquiv.refl}, σ (x : L) =
-      _
-  · rw [prod_singleton, AlgEquiv.coe_refl, id]
-  · rw [prod_sdiff <| subset_univ _, ← norm_eq_prod_automorphisms, coe_algebra_map_norm]
 #align ring_of_integers.is_unit_norm_of_is_galois RingOfIntegers.isUnit_norm_of_isGalois
 -/
 
 #print RingOfIntegers.dvd_norm /-
 /-- If `L/K` is a finite Galois extension of fields, then, for all `(x : 𝓞 L)` we have that
 `x ∣ algebra_map (𝓞 K) (𝓞 L) (norm K x)`. -/
-theorem dvd_norm [IsGalois K L] (x : 𝓞 L) : x ∣ algebraMap (𝓞 K) (𝓞 L) (norm K x) := by
-  classical
-  have hint : ∏ σ : L ≃ₐ[K] L in univ.erase AlgEquiv.refl, σ x ∈ 𝓞 L :=
-    Subalgebra.prod_mem _ fun σ hσ =>
-      (mem_ring_of_integers _ _).2 (IsIntegral.map σ (ring_of_integers.is_integral_coe x))
-  refine' ⟨⟨_, hint⟩, Subtype.ext _⟩
-  rw [coe_algebra_map_norm K x, norm_eq_prod_automorphisms]
-  simp [← Finset.mul_prod_erase _ _ (mem_univ AlgEquiv.refl)]
+theorem dvd_norm [IsGalois K L] (x : 𝓞 L) : x ∣ algebraMap (𝓞 K) (𝓞 L) (norm K x) := by classical
 #align ring_of_integers.dvd_norm RingOfIntegers.dvd_norm
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -73,7 +73,7 @@ theorem isUnit_norm_of_isGalois [IsGalois K L] {x : 𝓞 L} : IsUnit (norm K x)
   refine'
     @isUnit_of_mul_isUnit_right (𝓞 L) _
       ⟨(univ \ {AlgEquiv.refl}).Prod fun σ : L ≃ₐ[K] L => σ x,
-        prod_mem fun σ hσ => map_isIntegral (σ : L →+* L).toIntAlgHom x.2⟩
+        prod_mem fun σ hσ => IsIntegral.map (σ : L →+* L).toIntAlgHom x.2⟩
       _ _
   convert hx using 1
   ext
@@ -94,7 +94,7 @@ theorem dvd_norm [IsGalois K L] (x : 𝓞 L) : x ∣ algebraMap (𝓞 K) (𝓞 L
   classical
   have hint : ∏ σ : L ≃ₐ[K] L in univ.erase AlgEquiv.refl, σ x ∈ 𝓞 L :=
     Subalgebra.prod_mem _ fun σ hσ =>
-      (mem_ring_of_integers _ _).2 (map_isIntegral σ (ring_of_integers.is_integral_coe x))
+      (mem_ring_of_integers _ _).2 (IsIntegral.map σ (ring_of_integers.is_integral_coe x))
   refine' ⟨⟨_, hint⟩, Subtype.ext _⟩
   rw [coe_algebra_map_norm K x, norm_eq_prod_automorphisms]
   simp [← Finset.mul_prod_erase _ _ (mem_univ AlgEquiv.refl)]
@@ -119,7 +119,7 @@ theorem isUnit_norm [CharZero K] {x : 𝓞 F} : IsUnit (norm K x) ↔ IsUnit x :
   let L := normalClosure K F (AlgebraicClosureAux F)
   haveI : FiniteDimensional F L := FiniteDimensional.right K F L
   haveI : IsAlgClosure K (AlgebraicClosureAux F) :=
-    IsAlgClosure.ofAlgebraic K F (AlgebraicClosureAux F) (Algebra.isAlgebraic_of_finite K F)
+    IsAlgClosure.ofAlgebraic K F (AlgebraicClosureAux F) (Algebra.IsAlgebraic.of_finite K F)
   haveI : IsGalois F L := IsGalois.tower_top_of_isGalois K F L
   calc
     IsUnit (norm K x) ↔ IsUnit ((norm K) x ^ finrank F L) :=

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -3,8 +3,8 @@ Copyright (c) 2022 Riccardo Brasca. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Riccardo Brasca, Eric Rodriguez
 -/
-import Mathbin.NumberTheory.NumberField.Basic
-import Mathbin.RingTheory.Norm
+import NumberTheory.NumberField.Basic
+import RingTheory.Norm
 
 #align_import number_theory.number_field.norm from "leanprover-community/mathlib"@"1b089e3bdc3ce6b39cd472543474a0a137128c6c"
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/442a83d738cb208d3600056c489be16900ba701d

@@ -115,11 +115,11 @@ variable {F}
 #print RingOfIntegers.isUnit_norm /-
 theorem isUnit_norm [CharZero K] {x : 𝓞 F} : IsUnit (norm K x) ↔ IsUnit x :=
   by
-  letI : Algebra K (AlgebraicClosure K) := AlgebraicClosure.algebra K
-  let L := normalClosure K F (AlgebraicClosure F)
+  letI : Algebra K (AlgebraicClosureAux K) := AlgebraicClosureAux.algebra K
+  let L := normalClosure K F (AlgebraicClosureAux F)
   haveI : FiniteDimensional F L := FiniteDimensional.right K F L
-  haveI : IsAlgClosure K (AlgebraicClosure F) :=
-    IsAlgClosure.ofAlgebraic K F (AlgebraicClosure F) (Algebra.isAlgebraic_of_finite K F)
+  haveI : IsAlgClosure K (AlgebraicClosureAux F) :=
+    IsAlgClosure.ofAlgebraic K F (AlgebraicClosureAux F) (Algebra.isAlgebraic_of_finite K F)
   haveI : IsGalois F L := IsGalois.tower_top_of_isGalois K F L
   calc
     IsUnit (norm K x) ↔ IsUnit ((norm K) x ^ finrank F L) :=

mathlib commit https://github.com/leanprover-community/mathlib/commit/48a058d7e39a80ed56858505719a0b2197900999

@@ -41,7 +41,7 @@ noncomputable def norm [IsSeparable K L] : 𝓞 L →* 𝓞 K :=
 #align ring_of_integers.norm RingOfIntegers.norm
 -/
 
-attribute [local instance] NumberField.ringOfIntegersAlgebra
+attribute [local instance] NumberField.inst_ringOfIntegersAlgebra
 
 #print RingOfIntegers.coe_algebraMap_norm /-
 theorem coe_algebraMap_norm [IsSeparable K L] (x : 𝓞 L) :

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

@@ -2,15 +2,12 @@
 Copyright (c) 2022 Riccardo Brasca. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Riccardo Brasca, Eric Rodriguez
-
-! This file was ported from Lean 3 source module number_theory.number_field.norm
-! leanprover-community/mathlib commit 1b089e3bdc3ce6b39cd472543474a0a137128c6c
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.NumberTheory.NumberField.Basic
 import Mathbin.RingTheory.Norm
 
+#align_import number_theory.number_field.norm from "leanprover-community/mathlib"@"1b089e3bdc3ce6b39cd472543474a0a137128c6c"
+
 /-!
 # Norm in number fields
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/d30d31261cdb4d2f5e612eabc3c4bf45556350d5

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Riccardo Brasca, Eric Rodriguez
 
 ! This file was ported from Lean 3 source module number_theory.number_field.norm
-! leanprover-community/mathlib commit 00f91228655eecdcd3ac97a7fd8dbcb139fe990a
+! leanprover-community/mathlib commit 1b089e3bdc3ce6b39cd472543474a0a137128c6c
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -13,6 +13,9 @@ import Mathbin.RingTheory.Norm
 
 /-!
 # Norm in number fields
+
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
 Given a finite extension of number fields, we define the norm morphism as a function between the
 rings of integers.
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/f2ad3645af9effcdb587637dc28a6074edc813f9

@@ -33,30 +33,39 @@ namespace RingOfIntegers
 
 variable {L : Type _} (K : Type _) [Field K] [Field L] [Algebra K L] [FiniteDimensional K L]
 
+#print RingOfIntegers.norm /-
 /-- `algebra.norm` as a morphism betwen the rings of integers. -/
 @[simps]
 noncomputable def norm [IsSeparable K L] : 𝓞 L →* 𝓞 K :=
   ((Algebra.norm K).restrict (𝓞 L)).codRestrict (𝓞 K) fun x => isIntegral_norm K x.2
 #align ring_of_integers.norm RingOfIntegers.norm
+-/
 
 attribute [local instance] NumberField.ringOfIntegersAlgebra
 
+#print RingOfIntegers.coe_algebraMap_norm /-
 theorem coe_algebraMap_norm [IsSeparable K L] (x : 𝓞 L) :
     (algebraMap (𝓞 K) (𝓞 L) (norm K x) : L) = algebraMap K L (Algebra.norm K (x : L)) :=
   rfl
 #align ring_of_integers.coe_algebra_map_norm RingOfIntegers.coe_algebraMap_norm
+-/
 
+#print RingOfIntegers.coe_norm_algebraMap /-
 theorem coe_norm_algebraMap [IsSeparable K L] (x : 𝓞 K) :
     (norm K (algebraMap (𝓞 K) (𝓞 L) x) : K) = Algebra.norm K (algebraMap K L x) :=
   rfl
 #align ring_of_integers.coe_norm_algebra_map RingOfIntegers.coe_norm_algebraMap
+-/
 
+#print RingOfIntegers.norm_algebraMap /-
 theorem norm_algebraMap [IsSeparable K L] (x : 𝓞 K) :
     norm K (algebraMap (𝓞 K) (𝓞 L) x) = x ^ finrank K L := by
   rw [← Subtype.coe_inj, RingOfIntegers.coe_norm_algebraMap, Algebra.norm_algebraMap,
     SubsemiringClass.coe_pow]
 #align ring_of_integers.norm_algebra_map RingOfIntegers.norm_algebraMap
+-/
 
+#print RingOfIntegers.isUnit_norm_of_isGalois /-
 theorem isUnit_norm_of_isGalois [IsGalois K L] {x : 𝓞 L} : IsUnit (norm K x) ↔ IsUnit x := by
   classical
   refine' ⟨fun hx => _, IsUnit.map _⟩
@@ -76,7 +85,9 @@ theorem isUnit_norm_of_isGalois [IsGalois K L] {x : 𝓞 L} : IsUnit (norm K x)
   · rw [prod_singleton, AlgEquiv.coe_refl, id]
   · rw [prod_sdiff <| subset_univ _, ← norm_eq_prod_automorphisms, coe_algebra_map_norm]
 #align ring_of_integers.is_unit_norm_of_is_galois RingOfIntegers.isUnit_norm_of_isGalois
+-/
 
+#print RingOfIntegers.dvd_norm /-
 /-- If `L/K` is a finite Galois extension of fields, then, for all `(x : 𝓞 L)` we have that
 `x ∣ algebra_map (𝓞 K) (𝓞 L) (norm K x)`. -/
 theorem dvd_norm [IsGalois K L] (x : 𝓞 L) : x ∣ algebraMap (𝓞 K) (𝓞 L) (norm K x) := by
@@ -88,16 +99,20 @@ theorem dvd_norm [IsGalois K L] (x : 𝓞 L) : x ∣ algebraMap (𝓞 K) (𝓞 L
   rw [coe_algebra_map_norm K x, norm_eq_prod_automorphisms]
   simp [← Finset.mul_prod_erase _ _ (mem_univ AlgEquiv.refl)]
 #align ring_of_integers.dvd_norm RingOfIntegers.dvd_norm
+-/
 
 variable (F : Type _) [Field F] [Algebra K F] [IsSeparable K F] [FiniteDimensional K F]
 
+#print RingOfIntegers.norm_norm /-
 theorem norm_norm [IsSeparable K L] [Algebra F L] [IsSeparable F L] [FiniteDimensional F L]
     [IsScalarTower K F L] (x : 𝓞 L) : norm K (norm F x) = norm K x := by
   rw [← Subtype.coe_inj, norm_apply_coe, norm_apply_coe, norm_apply_coe, Algebra.norm_norm]
 #align ring_of_integers.norm_norm RingOfIntegers.norm_norm
+-/
 
 variable {F}
 
+#print RingOfIntegers.isUnit_norm /-
 theorem isUnit_norm [CharZero K] {x : 𝓞 F} : IsUnit (norm K x) ↔ IsUnit x :=
   by
   letI : Algebra K (AlgebraicClosure K) := AlgebraicClosure.algebra K
@@ -116,6 +131,7 @@ theorem isUnit_norm [CharZero K] {x : 𝓞 F} : IsUnit (norm K x) ↔ IsUnit x :
     _ ↔ IsUnit (x ^ finrank F L) := (congr_arg IsUnit (norm_algebraMap F _)).to_iff
     _ ↔ IsUnit x := isUnit_pow_iff (pos_iff_ne_zero.mp finrank_pos)
 #align ring_of_integers.is_unit_norm RingOfIntegers.isUnit_norm
+-/
 
 end RingOfIntegers
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/a3e83f0fa4391c8740f7d773a7a9b74e311ae2a3

@@ -70,8 +70,8 @@ theorem isUnit_norm_of_isGalois [IsGalois K L] {x : 𝓞 L} : IsUnit (norm K x)
   ext
   push_cast
   convert_to
-    (((univ \ {AlgEquiv.refl}).Prod fun σ : L ≃ₐ[K] L => σ x) *
-        ∏ σ : L ≃ₐ[K] L in {AlgEquiv.refl}, σ (x : L)) =
+    ((univ \ {AlgEquiv.refl}).Prod fun σ : L ≃ₐ[K] L => σ x) *
+        ∏ σ : L ≃ₐ[K] L in {AlgEquiv.refl}, σ (x : L) =
       _
   · rw [prod_singleton, AlgEquiv.coe_refl, id]
   · rw [prod_sdiff <| subset_univ _, ← norm_eq_prod_automorphisms, coe_algebra_map_norm]
@@ -81,7 +81,7 @@ theorem isUnit_norm_of_isGalois [IsGalois K L] {x : 𝓞 L} : IsUnit (norm K x)
 `x ∣ algebra_map (𝓞 K) (𝓞 L) (norm K x)`. -/
 theorem dvd_norm [IsGalois K L] (x : 𝓞 L) : x ∣ algebraMap (𝓞 K) (𝓞 L) (norm K x) := by
   classical
-  have hint : (∏ σ : L ≃ₐ[K] L in univ.erase AlgEquiv.refl, σ x) ∈ 𝓞 L :=
+  have hint : ∏ σ : L ≃ₐ[K] L in univ.erase AlgEquiv.refl, σ x ∈ 𝓞 L :=
     Subalgebra.prod_mem _ fun σ hσ =>
       (mem_ring_of_integers _ _).2 (map_isIntegral σ (ring_of_integers.is_integral_coe x))
   refine' ⟨⟨_, hint⟩, Subtype.ext _⟩

mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0

@@ -115,7 +115,6 @@ theorem isUnit_norm [CharZero K] {x : 𝓞 F} : IsUnit (norm K x) ↔ IsUnit x :
     _ ↔ IsUnit (norm F (algebraMap (𝓞 F) (𝓞 L) x)) := (is_unit_norm_of_is_galois F).symm
     _ ↔ IsUnit (x ^ finrank F L) := (congr_arg IsUnit (norm_algebraMap F _)).to_iff
     _ ↔ IsUnit x := isUnit_pow_iff (pos_iff_ne_zero.mp finrank_pos)
-    
 #align ring_of_integers.is_unit_norm RingOfIntegers.isUnit_norm
 
 end RingOfIntegers

mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297

@@ -59,33 +59,34 @@ theorem norm_algebraMap [IsSeparable K L] (x : 𝓞 K) :
 
 theorem isUnit_norm_of_isGalois [IsGalois K L] {x : 𝓞 L} : IsUnit (norm K x) ↔ IsUnit x := by
   classical
-    refine' ⟨fun hx => _, IsUnit.map _⟩
-    replace hx : IsUnit (algebraMap (𝓞 K) (𝓞 L) <| norm K x) := hx.map (algebraMap (𝓞 K) <| 𝓞 L)
-    refine'
-      @isUnit_of_mul_isUnit_right (𝓞 L) _
-        ⟨(univ \ {AlgEquiv.refl}).Prod fun σ : L ≃ₐ[K] L => σ x,
-          prod_mem fun σ hσ => map_isIntegral (σ : L →+* L).toIntAlgHom x.2⟩
-        _ _
-    convert hx using 1
-    ext
-    push_cast
-    convert_to(((univ \ {AlgEquiv.refl}).Prod fun σ : L ≃ₐ[K] L => σ x) *
-          ∏ σ : L ≃ₐ[K] L in {AlgEquiv.refl}, σ (x : L)) =
-        _
-    · rw [prod_singleton, AlgEquiv.coe_refl, id]
-    · rw [prod_sdiff <| subset_univ _, ← norm_eq_prod_automorphisms, coe_algebra_map_norm]
+  refine' ⟨fun hx => _, IsUnit.map _⟩
+  replace hx : IsUnit (algebraMap (𝓞 K) (𝓞 L) <| norm K x) := hx.map (algebraMap (𝓞 K) <| 𝓞 L)
+  refine'
+    @isUnit_of_mul_isUnit_right (𝓞 L) _
+      ⟨(univ \ {AlgEquiv.refl}).Prod fun σ : L ≃ₐ[K] L => σ x,
+        prod_mem fun σ hσ => map_isIntegral (σ : L →+* L).toIntAlgHom x.2⟩
+      _ _
+  convert hx using 1
+  ext
+  push_cast
+  convert_to
+    (((univ \ {AlgEquiv.refl}).Prod fun σ : L ≃ₐ[K] L => σ x) *
+        ∏ σ : L ≃ₐ[K] L in {AlgEquiv.refl}, σ (x : L)) =
+      _
+  · rw [prod_singleton, AlgEquiv.coe_refl, id]
+  · rw [prod_sdiff <| subset_univ _, ← norm_eq_prod_automorphisms, coe_algebra_map_norm]
 #align ring_of_integers.is_unit_norm_of_is_galois RingOfIntegers.isUnit_norm_of_isGalois
 
 /-- If `L/K` is a finite Galois extension of fields, then, for all `(x : 𝓞 L)` we have that
 `x ∣ algebra_map (𝓞 K) (𝓞 L) (norm K x)`. -/
 theorem dvd_norm [IsGalois K L] (x : 𝓞 L) : x ∣ algebraMap (𝓞 K) (𝓞 L) (norm K x) := by
   classical
-    have hint : (∏ σ : L ≃ₐ[K] L in univ.erase AlgEquiv.refl, σ x) ∈ 𝓞 L :=
-      Subalgebra.prod_mem _ fun σ hσ =>
-        (mem_ring_of_integers _ _).2 (map_isIntegral σ (ring_of_integers.is_integral_coe x))
-    refine' ⟨⟨_, hint⟩, Subtype.ext _⟩
-    rw [coe_algebra_map_norm K x, norm_eq_prod_automorphisms]
-    simp [← Finset.mul_prod_erase _ _ (mem_univ AlgEquiv.refl)]
+  have hint : (∏ σ : L ≃ₐ[K] L in univ.erase AlgEquiv.refl, σ x) ∈ 𝓞 L :=
+    Subalgebra.prod_mem _ fun σ hσ =>
+      (mem_ring_of_integers _ _).2 (map_isIntegral σ (ring_of_integers.is_integral_coe x))
+  refine' ⟨⟨_, hint⟩, Subtype.ext _⟩
+  rw [coe_algebra_map_norm K x, norm_eq_prod_automorphisms]
+  simp [← Finset.mul_prod_erase _ _ (mem_univ AlgEquiv.refl)]
 #align ring_of_integers.dvd_norm RingOfIntegers.dvd_norm
 
 variable (F : Type _) [Field F] [Algebra K F] [IsSeparable K F] [FiniteDimensional K F]

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -25,7 +25,7 @@ rings of integers.
 -/
 
 
-open NumberField BigOperators
+open scoped NumberField BigOperators
 
 open Finset NumberField Algebra FiniteDimensional
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/3905fa80e62c0898131285baab35559fbc4e5cda

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Riccardo Brasca, Eric Rodriguez
 
 ! This file was ported from Lean 3 source module number_theory.number_field.norm
-! leanprover-community/mathlib commit ee0c179cd3c8a45aa5bffbf1b41d8dbede452865
+! leanprover-community/mathlib commit 00f91228655eecdcd3ac97a7fd8dbcb139fe990a
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -27,7 +27,7 @@ rings of integers.
 
 open NumberField BigOperators
 
-open Finset NumberField Algebra
+open Finset NumberField Algebra FiniteDimensional
 
 namespace RingOfIntegers
 
@@ -46,7 +46,18 @@ theorem coe_algebraMap_norm [IsSeparable K L] (x : 𝓞 L) :
   rfl
 #align ring_of_integers.coe_algebra_map_norm RingOfIntegers.coe_algebraMap_norm
 
-theorem isUnit_norm [IsGalois K L] {x : 𝓞 L} : IsUnit (norm K x) ↔ IsUnit x := by
+theorem coe_norm_algebraMap [IsSeparable K L] (x : 𝓞 K) :
+    (norm K (algebraMap (𝓞 K) (𝓞 L) x) : K) = Algebra.norm K (algebraMap K L x) :=
+  rfl
+#align ring_of_integers.coe_norm_algebra_map RingOfIntegers.coe_norm_algebraMap
+
+theorem norm_algebraMap [IsSeparable K L] (x : 𝓞 K) :
+    norm K (algebraMap (𝓞 K) (𝓞 L) x) = x ^ finrank K L := by
+  rw [← Subtype.coe_inj, RingOfIntegers.coe_norm_algebraMap, Algebra.norm_algebraMap,
+    SubsemiringClass.coe_pow]
+#align ring_of_integers.norm_algebra_map RingOfIntegers.norm_algebraMap
+
+theorem isUnit_norm_of_isGalois [IsGalois K L] {x : 𝓞 L} : IsUnit (norm K x) ↔ IsUnit x := by
   classical
     refine' ⟨fun hx => _, IsUnit.map _⟩
     replace hx : IsUnit (algebraMap (𝓞 K) (𝓞 L) <| norm K x) := hx.map (algebraMap (𝓞 K) <| 𝓞 L)
@@ -63,7 +74,7 @@ theorem isUnit_norm [IsGalois K L] {x : 𝓞 L} : IsUnit (norm K x) ↔ IsUnit x
         _
     · rw [prod_singleton, AlgEquiv.coe_refl, id]
     · rw [prod_sdiff <| subset_univ _, ← norm_eq_prod_automorphisms, coe_algebra_map_norm]
-#align ring_of_integers.is_unit_norm RingOfIntegers.isUnit_norm
+#align ring_of_integers.is_unit_norm_of_is_galois RingOfIntegers.isUnit_norm_of_isGalois
 
 /-- If `L/K` is a finite Galois extension of fields, then, for all `(x : 𝓞 L)` we have that
 `x ∣ algebra_map (𝓞 K) (𝓞 L) (norm K x)`. -/
@@ -77,5 +88,34 @@ theorem dvd_norm [IsGalois K L] (x : 𝓞 L) : x ∣ algebraMap (𝓞 K) (𝓞 L
     simp [← Finset.mul_prod_erase _ _ (mem_univ AlgEquiv.refl)]
 #align ring_of_integers.dvd_norm RingOfIntegers.dvd_norm
 
+variable (F : Type _) [Field F] [Algebra K F] [IsSeparable K F] [FiniteDimensional K F]
+
+theorem norm_norm [IsSeparable K L] [Algebra F L] [IsSeparable F L] [FiniteDimensional F L]
+    [IsScalarTower K F L] (x : 𝓞 L) : norm K (norm F x) = norm K x := by
+  rw [← Subtype.coe_inj, norm_apply_coe, norm_apply_coe, norm_apply_coe, Algebra.norm_norm]
+#align ring_of_integers.norm_norm RingOfIntegers.norm_norm
+
+variable {F}
+
+theorem isUnit_norm [CharZero K] {x : 𝓞 F} : IsUnit (norm K x) ↔ IsUnit x :=
+  by
+  letI : Algebra K (AlgebraicClosure K) := AlgebraicClosure.algebra K
+  let L := normalClosure K F (AlgebraicClosure F)
+  haveI : FiniteDimensional F L := FiniteDimensional.right K F L
+  haveI : IsAlgClosure K (AlgebraicClosure F) :=
+    IsAlgClosure.ofAlgebraic K F (AlgebraicClosure F) (Algebra.isAlgebraic_of_finite K F)
+  haveI : IsGalois F L := IsGalois.tower_top_of_isGalois K F L
+  calc
+    IsUnit (norm K x) ↔ IsUnit ((norm K) x ^ finrank F L) :=
+      (isUnit_pow_iff (pos_iff_ne_zero.mp finrank_pos)).symm
+    _ ↔ IsUnit (norm K (algebraMap (𝓞 F) (𝓞 L) x)) := by
+      rw [← norm_norm K F (algebraMap (𝓞 F) (𝓞 L) x), norm_algebraMap F _, map_pow]
+    _ ↔ IsUnit (algebraMap (𝓞 F) (𝓞 L) x) := (is_unit_norm_of_is_galois K)
+    _ ↔ IsUnit (norm F (algebraMap (𝓞 F) (𝓞 L) x)) := (is_unit_norm_of_is_galois F).symm
+    _ ↔ IsUnit (x ^ finrank F L) := (congr_arg IsUnit (norm_algebraMap F _)).to_iff
+    _ ↔ IsUnit x := isUnit_pow_iff (pos_iff_ne_zero.mp finrank_pos)
+    
+#align ring_of_integers.is_unit_norm RingOfIntegers.isUnit_norm
+
 end RingOfIntegers
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce7e9d53d4bbc38065db3b595cd5bd73c323bc1d

@@ -58,8 +58,7 @@ theorem isUnit_norm [IsGalois K L] {x : 𝓞 L} : IsUnit (norm K x) ↔ IsUnit x
     convert hx using 1
     ext
     push_cast
-    convert_to
-      (((univ \ {AlgEquiv.refl}).Prod fun σ : L ≃ₐ[K] L => σ x) *
+    convert_to(((univ \ {AlgEquiv.refl}).Prod fun σ : L ≃ₐ[K] L => σ x) *
           ∏ σ : L ≃ₐ[K] L in {AlgEquiv.refl}, σ (x : L)) =
         _
     · rw [prod_singleton, AlgEquiv.coe_refl, id]

mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc

Co-authored-by: Moritz Firsching <firsching@google.com>

@@ -113,7 +113,7 @@ theorem isUnit_norm [CharZero K] {x : 𝓞 F} : IsUnit (norm K x) ↔ IsUnit x :
       (isUnit_pow_iff (pos_iff_ne_zero.mp finrank_pos)).symm
     _ ↔ IsUnit (norm K (algebraMap (𝓞 F) (𝓞 L) x)) := by
       rw [← norm_norm K F (algebraMap (𝓞 F) (𝓞 L) x), norm_algebraMap F _, map_pow]
-    _ ↔ IsUnit (algebraMap (𝓞 F) (𝓞 L) x) := (isUnit_norm_of_isGalois K)
+    _ ↔ IsUnit (algebraMap (𝓞 F) (𝓞 L) x) := isUnit_norm_of_isGalois K
     _ ↔ IsUnit (norm F (algebraMap (𝓞 F) (𝓞 L) x)) := (isUnit_norm_of_isGalois F).symm
     _ ↔ IsUnit (x ^ finrank F L) := (congr_arg IsUnit (norm_algebraMap F _)).to_iff
     _ ↔ IsUnit x := isUnit_pow_iff (pos_iff_ne_zero.mp finrank_pos)

With multiple changes, it is a good time to check if existing set_option maxHeartbeats and set_option synthInstance.maxHeartbeats remain necessary. This brings the number of files with such down from 23 to 9. Most are straight deletions though I did change one proof.

@@ -101,7 +101,6 @@ theorem norm_norm [IsSeparable K L] [Algebra F L] [IsSeparable F L] [FiniteDimen
 
 variable {F}
 
-set_option synthInstance.maxHeartbeats 60000 in
 theorem isUnit_norm [CharZero K] {x : 𝓞 F} : IsUnit (norm K x) ↔ IsUnit x := by
   letI : Algebra K (AlgebraicClosure K) := AlgebraicClosure.instAlgebra K
   let L := normalClosure K F (AlgebraicClosure F)

Those lemmas were very recently added in #9397. Also make them iffs and golf.

Part of #9411

@@ -101,6 +101,7 @@ theorem norm_norm [IsSeparable K L] [Algebra F L] [IsSeparable F L] [FiniteDimen
 
 variable {F}
 
+set_option synthInstance.maxHeartbeats 60000 in
 theorem isUnit_norm [CharZero K] {x : 𝓞 F} : IsUnit (norm K x) ↔ IsUnit x := by
   letI : Algebra K (AlgebraicClosure K) := AlgebraicClosure.instAlgebra K
   let L := normalClosure K F (AlgebraicClosure F)

From flt-regular.

@@ -26,13 +26,17 @@ open scoped NumberField BigOperators
 
 open Finset NumberField Algebra FiniteDimensional
 
-section rat
+section Rat
 
-theorem Algebra.coe_norm_int {K : Type*} [Field K] [NumberField K] (x : 𝓞 K) :
-    Algebra.norm ℤ x = Algebra.norm ℚ (x : K) :=
+variable {K : Type*} [Field K] [NumberField K] (x : 𝓞 K)
+
+theorem Algebra.coe_norm_int : (Algebra.norm ℤ x : ℚ) = Algebra.norm ℚ (x : K) :=
   (Algebra.norm_localization (R := ℤ) (Rₘ := ℚ) (S := 𝓞 K) (Sₘ := K) (nonZeroDivisors ℤ) x).symm
 
-end rat
+theorem Algebra.coe_trace_int : (Algebra.trace ℤ _ x : ℚ) = Algebra.trace ℚ K x :=
+  (Algebra.trace_localization (R := ℤ) (Rₘ := ℚ) (S := 𝓞 K) (Sₘ := K) (nonZeroDivisors ℤ) x).symm
+
+end Rat
 
 namespace RingOfIntegers
 

The following lemma:

theorem Algebra.coe_norm_int {K : Type*} [Field K] [NumberField K] (x : 𝓞 K) :
    Algebra.norm ℤ x = Algebra.norm ℚ (x : K)

is currently in NumberField.Units but it belongs to NumberField.Norm

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Riccardo Brasca, Eric Rodriguez
 -/
 import Mathlib.NumberTheory.NumberField.Basic
-import Mathlib.RingTheory.Norm
+import Mathlib.RingTheory.Localization.NormTrace
 
 #align_import number_theory.number_field.norm from "leanprover-community/mathlib"@"00f91228655eecdcd3ac97a7fd8dbcb139fe990a"
 
@@ -26,6 +26,14 @@ open scoped NumberField BigOperators
 
 open Finset NumberField Algebra FiniteDimensional
 
+section rat
+
+theorem Algebra.coe_norm_int {K : Type*} [Field K] [NumberField K] (x : 𝓞 K) :
+    Algebra.norm ℤ x = Algebra.norm ℚ (x : K) :=
+  (Algebra.norm_localization (R := ℤ) (Rₘ := ℚ) (S := 𝓞 K) (Sₘ := K) (nonZeroDivisors ℤ) x).symm
+
+end rat
+
 namespace RingOfIntegers
 
 variable {L : Type*} (K : Type*) [Field K] [Field L] [Algebra K L] [FiniteDimensional K L]

Zulip

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.IsIntegral which makes it unfold to IsIntegral rather than RingHom.IsIntegralElem because the former has much more APIs.

• Fix lemma names involving is_integral which are actually about IsIntegralElem: RingHom.is_integral_map → RingHom.isIntegralElem_map RingHom.is_integral_of_mem_closure → RingHom.IsIntegralElem.of_mem_closure RingHom.is_integral_zero/one → RingHom.isIntegralElem_zero/one RingHom.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_)noetherian down and golf them.

• Remove (Algebra.)isIntegral_of_finite that work only over fields, in favor of the more general (Algebra.)isIntegral.of_finite.

• Merge duplicate lemmas isIntegral_of_isScalarTower and isIntegral_tower_top_of_isIntegral into IsIntegral.tower_top.

• Golf IsIntegral.of_mem_of_fg by first proving IsIntegral.of_finite using 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.tmul and isField_of_isIntegral_of_isField(').

• Combine isIntegral_trans_aux into isIntegral_trans and golf.

• Add Algebra namespace to isIntegral_sup.

• rename lemmas for dot notation: RingHom.isIntegral_trans → RingHom.IsIntegral.trans RingHom.isIntegral_quotient/tower_bot/top_of_isIntegral → RingHom.IsIntegral.quotient/tower_bot/top isIntegral_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 with IsIntegral) Algebra.isAlgebraic_of_finite → Algebra.IsAlgebraic.of_finite Algebra.isAlgebraic_trans → Algebra.IsAlgebraic.trans

• Add new lemmasAlgebra.IsIntegral.isAlgebraic, isAlgebraic_algHom_iff, and Algebra.IsAlgebraic.of_injective to 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>

@@ -58,7 +58,7 @@ theorem isUnit_norm_of_isGalois [IsGalois K L] {x : 𝓞 L} : IsUnit (norm K x)
   replace hx : IsUnit (algebraMap (𝓞 K) (𝓞 L) <| norm K x) := hx.map (algebraMap (𝓞 K) <| 𝓞 L)
   refine' @isUnit_of_mul_isUnit_right (𝓞 L) _
     ⟨(univ \ {AlgEquiv.refl}).prod fun σ : L ≃ₐ[K] L => σ x,
-      prod_mem fun σ _ => IsIntegral.map (σ : L →+* L).toIntAlgHom x.2⟩ _ _
+      prod_mem fun σ _ => x.2.map (σ : L →+* L).toIntAlgHom⟩ _ _
   convert hx using 1
   ext
   push_cast
@@ -74,7 +74,7 @@ theorem dvd_norm [IsGalois K L] (x : 𝓞 L) : x ∣ algebraMap (𝓞 K) (𝓞 L
   classical
   have hint : ∏ σ : L ≃ₐ[K] L in univ.erase AlgEquiv.refl, σ x ∈ 𝓞 L :=
     Subalgebra.prod_mem _ fun σ _ =>
-      (mem_ringOfIntegers _ _).2 (IsIntegral.map σ (RingOfIntegers.isIntegral_coe x))
+      (mem_ringOfIntegers _ _).2 ((RingOfIntegers.isIntegral_coe x).map σ)
   refine' ⟨⟨_, hint⟩, Subtype.ext _⟩
   rw [coe_algebraMap_norm K x, norm_eq_prod_automorphisms]
   simp [← Finset.mul_prod_erase _ _ (mem_univ AlgEquiv.refl)]
@@ -94,7 +94,7 @@ theorem isUnit_norm [CharZero K] {x : 𝓞 F} : IsUnit (norm K x) ↔ IsUnit x :
   let L := normalClosure K F (AlgebraicClosure F)
   haveI : FiniteDimensional F L := FiniteDimensional.right K F L
   haveI : IsAlgClosure K (AlgebraicClosure F) :=
-    IsAlgClosure.ofAlgebraic K F (AlgebraicClosure F) (Algebra.isAlgebraic_of_finite K F)
+    IsAlgClosure.ofAlgebraic K F (AlgebraicClosure F) (Algebra.IsAlgebraic.of_finite K F)
   haveI : IsGalois F L := IsGalois.tower_top_of_isGalois K F L
   calc
     IsUnit (norm K x) ↔ IsUnit ((norm K) x ^ finrank F L) :=

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 |

@@ -58,7 +58,7 @@ theorem isUnit_norm_of_isGalois [IsGalois K L] {x : 𝓞 L} : IsUnit (norm K x)
   replace hx : IsUnit (algebraMap (𝓞 K) (𝓞 L) <| norm K x) := hx.map (algebraMap (𝓞 K) <| 𝓞 L)
   refine' @isUnit_of_mul_isUnit_right (𝓞 L) _
     ⟨(univ \ {AlgEquiv.refl}).prod fun σ : L ≃ₐ[K] L => σ x,
-      prod_mem fun σ _ => map_isIntegral (σ : L →+* L).toIntAlgHom x.2⟩ _ _
+      prod_mem fun σ _ => IsIntegral.map (σ : L →+* L).toIntAlgHom x.2⟩ _ _
   convert hx using 1
   ext
   push_cast
@@ -74,7 +74,7 @@ theorem dvd_norm [IsGalois K L] (x : 𝓞 L) : x ∣ algebraMap (𝓞 K) (𝓞 L
   classical
   have hint : ∏ σ : L ≃ₐ[K] L in univ.erase AlgEquiv.refl, σ x ∈ 𝓞 L :=
     Subalgebra.prod_mem _ fun σ _ =>
-      (mem_ringOfIntegers _ _).2 (map_isIntegral σ (RingOfIntegers.isIntegral_coe x))
+      (mem_ringOfIntegers _ _).2 (IsIntegral.map σ (RingOfIntegers.isIntegral_coe x))
   refine' ⟨⟨_, hint⟩, Subtype.ext _⟩
   rw [coe_algebraMap_norm K x, norm_eq_prod_automorphisms]
   simp [← Finset.mul_prod_erase _ _ (mem_univ AlgEquiv.refl)]

We remove all possible occurences of Type _ and Sort _ in favor of Type* and Sort*.

This has nice performance benefits.

@@ -28,7 +28,7 @@ open Finset NumberField Algebra FiniteDimensional
 
 namespace RingOfIntegers
 
-variable {L : Type _} (K : Type _) [Field K] [Field L] [Algebra K L] [FiniteDimensional K L]
+variable {L : Type*} (K : Type*) [Field K] [Field L] [Algebra K L] [FiniteDimensional K L]
 
 /-- `Algebra.norm` as a morphism betwen the rings of integers. -/
 @[simps!]
@@ -80,7 +80,7 @@ theorem dvd_norm [IsGalois K L] (x : 𝓞 L) : x ∣ algebraMap (𝓞 K) (𝓞 L
   simp [← Finset.mul_prod_erase _ _ (mem_univ AlgEquiv.refl)]
 #align ring_of_integers.dvd_norm RingOfIntegers.dvd_norm
 
-variable (F : Type _) [Field F] [Algebra K F] [IsSeparable K F] [FiniteDimensional K F]
+variable (F : Type*) [Field F] [Algebra K F] [IsSeparable K F] [FiniteDimensional K F]
 
 theorem norm_norm [IsSeparable K L] [Algebra F L] [IsSeparable F L] [FiniteDimensional F L]
     [IsScalarTower K F L] (x : 𝓞 L) : norm K (norm F x) = norm K x := by

Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -36,8 +36,6 @@ noncomputable def norm [IsSeparable K L] : 𝓞 L →* 𝓞 K :=
   ((Algebra.norm K).restrict (𝓞 L)).codRestrict (𝓞 K) fun x => isIntegral_norm K x.2
 #align ring_of_integers.norm RingOfIntegers.norm
 
-attribute [local instance] NumberField.ringOfIntegersAlgebra
-
 theorem coe_algebraMap_norm [IsSeparable K L] (x : 𝓞 L) :
     (algebraMap (𝓞 K) (𝓞 L) (norm K x) : L) = algebraMap K L (Algebra.norm K (x : L)) :=
   rfl

• depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -2,15 +2,12 @@
 Copyright (c) 2022 Riccardo Brasca. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Riccardo Brasca, Eric Rodriguez
-
-! This file was ported from Lean 3 source module number_theory.number_field.norm
-! leanprover-community/mathlib commit 00f91228655eecdcd3ac97a7fd8dbcb139fe990a
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.NumberTheory.NumberField.Basic
 import Mathlib.RingTheory.Norm
 
+#align_import number_theory.number_field.norm from "leanprover-community/mathlib"@"00f91228655eecdcd3ac97a7fd8dbcb139fe990a"
+
 /-!
 # Norm in number fields
 Given a finite extension of number fields, we define the norm morphism as a function between the