field_theory.galois ⟷ Mathlib.FieldTheory.Galois

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

@@ -470,7 +470,7 @@ theorem of_separable_splitting_field_aux [hFE : FiniteDimensional F E] [sp : p.I
   by
   have h : IsIntegral K x :=
     IsIntegral.tower_top (isIntegral_of_noetherian (IsNoetherian.iff_fg.2 hFE) x)
-  have h1 : p ≠ 0 := fun hp => by rwa [hp, Polynomial.map_zero, Polynomial.roots_zero] at hx 
+  have h1 : p ≠ 0 := fun hp => by rwa [hp, Polynomial.map_zero, Polynomial.roots_zero] at hx
   have h2 : minpoly K x ∣ p.map (algebraMap F K) :=
     by
     apply minpoly.dvd
@@ -514,7 +514,7 @@ theorem of_separable_splitting_field [sp : p.IsSplittingField F E] (hp : p.Separ
   let P : IntermediateField F E → Prop := fun K => Fintype.card (K →ₐ[F] E) = finrank F K
   suffices P (IntermediateField.adjoin F ↑s)
     by
-    rw [AdjoinRoot] at this 
+    rw [AdjoinRoot] at this
     apply of_card_aut_eq_finrank
     rw [← Eq.trans this (LinearEquiv.finrank_eq intermediate_field.top_equiv.to_linear_equiv)]
     exact
@@ -525,9 +525,9 @@ theorem of_separable_splitting_field [sp : p.IsSplittingField F E] (hp : p.Separ
   · have key :=
       IntermediateField.card_algHom_adjoin_integral F
         (show IsIntegral F (0 : E) from isIntegral_zero)
-    rw [minpoly.zero, Polynomial.natDegree_X] at key 
+    rw [minpoly.zero, Polynomial.natDegree_X] at key
     specialize key Polynomial.separable_X (Polynomial.splits_X (algebraMap F E))
-    rw [← @Subalgebra.finrank_bot F E _ _ _, ← IntermediateField.bot_toSubalgebra] at key 
+    rw [← @Subalgebra.finrank_bot F E _ _ _, ← IntermediateField.bot_toSubalgebra] at key
     refine' Eq.trans _ key
     apply Fintype.card_congr
     rw [IntermediateField.adjoin_zero]

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

@@ -286,6 +286,16 @@ theorem fixingSubgroup_fixedField [FiniteDimensional F E] : fixingSubgroup (fixe
   by
   have H_le : H ≤ fixingSubgroup (fixed_field H) := (le_iff_le _ _).mp le_rfl
   classical
+  suffices Fintype.card H = Fintype.card (fixingSubgroup (fixed_field H)) by
+    exact
+      SetLike.coe_injective
+        (Set.eq_of_inclusion_surjective
+            ((Fintype.bijective_iff_injective_and_card (Set.inclusion H_le)).mpr
+                ⟨Set.inclusion_injective H_le, this⟩).2).symm
+  apply Fintype.card_congr
+  refine' (FixedPoints.toAlgHomEquiv H E).trans _
+  refine' (algEquivEquivAlgHom (fixed_field H) E).toEquiv.symm.trans _
+  exact (fixing_subgroup_equiv (fixed_field H)).toEquiv.symm
 #align intermediate_field.fixing_subgroup_fixed_field IntermediateField.fixingSubgroup_fixedField
 -/
 
@@ -325,6 +335,9 @@ theorem fixedField_fixingSubgroup [FiniteDimensional F E] [h : IsGalois F E] :
     finrank K E = finrank (IntermediateField.fixedField (IntermediateField.fixingSubgroup K)) E by
     exact (IntermediateField.eq_of_le_of_finrank_eq' K_le this).symm
   classical
+  rw [IntermediateField.finrank_fixedField_eq_card,
+    Fintype.card_congr (IntermediateField.fixingSubgroupEquiv K).toEquiv]
+  exact (card_aut_eq_finrank K E).symm
 #align is_galois.fixed_field_fixing_subgroup IsGalois.fixedField_fixingSubgroup
 -/
 
@@ -415,7 +428,7 @@ theorem of_fixedField_eq_bot [FiniteDimensional F E]
     (h : IntermediateField.fixedField (⊤ : Subgroup (E ≃ₐ[F] E)) = ⊥) : IsGalois F E :=
   by
   rw [← isGalois_iff_isGalois_bot, ← h]
-  classical
+  classical exact IsGalois.of_fixed_field E (⊤ : Subgroup (E ≃ₐ[F] E))
 #align is_galois.of_fixed_field_eq_bot IsGalois.of_fixedField_eq_bot
 -/
 
@@ -426,6 +439,14 @@ theorem of_card_aut_eq_finrank [FiniteDimensional F E]
   apply of_fixed_field_eq_bot
   have p : 0 < finrank (IntermediateField.fixedField (⊤ : Subgroup (E ≃ₐ[F] E))) E := finrank_pos
   classical
+  rw [← IntermediateField.finrank_eq_one_iff, ← mul_left_inj' (ne_of_lt p).symm,
+    finrank_mul_finrank, ← h, one_mul, IntermediateField.finrank_fixedField_eq_card]
+  apply Fintype.card_congr
+  exact
+    { toFun := fun g => ⟨g, Subgroup.mem_top g⟩
+      invFun := coe
+      left_inv := fun g => rfl
+      right_inv := fun _ => by ext; rfl }
 #align is_galois.of_card_aut_eq_finrank IsGalois.of_card_aut_eq_finrank
 -/
 

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

@@ -286,16 +286,6 @@ theorem fixingSubgroup_fixedField [FiniteDimensional F E] : fixingSubgroup (fixe
   by
   have H_le : H ≤ fixingSubgroup (fixed_field H) := (le_iff_le _ _).mp le_rfl
   classical
-  suffices Fintype.card H = Fintype.card (fixingSubgroup (fixed_field H)) by
-    exact
-      SetLike.coe_injective
-        (Set.eq_of_inclusion_surjective
-            ((Fintype.bijective_iff_injective_and_card (Set.inclusion H_le)).mpr
-                ⟨Set.inclusion_injective H_le, this⟩).2).symm
-  apply Fintype.card_congr
-  refine' (FixedPoints.toAlgHomEquiv H E).trans _
-  refine' (algEquivEquivAlgHom (fixed_field H) E).toEquiv.symm.trans _
-  exact (fixing_subgroup_equiv (fixed_field H)).toEquiv.symm
 #align intermediate_field.fixing_subgroup_fixed_field IntermediateField.fixingSubgroup_fixedField
 -/
 
@@ -335,9 +325,6 @@ theorem fixedField_fixingSubgroup [FiniteDimensional F E] [h : IsGalois F E] :
     finrank K E = finrank (IntermediateField.fixedField (IntermediateField.fixingSubgroup K)) E by
     exact (IntermediateField.eq_of_le_of_finrank_eq' K_le this).symm
   classical
-  rw [IntermediateField.finrank_fixedField_eq_card,
-    Fintype.card_congr (IntermediateField.fixingSubgroupEquiv K).toEquiv]
-  exact (card_aut_eq_finrank K E).symm
 #align is_galois.fixed_field_fixing_subgroup IsGalois.fixedField_fixingSubgroup
 -/
 
@@ -428,7 +415,7 @@ theorem of_fixedField_eq_bot [FiniteDimensional F E]
     (h : IntermediateField.fixedField (⊤ : Subgroup (E ≃ₐ[F] E)) = ⊥) : IsGalois F E :=
   by
   rw [← isGalois_iff_isGalois_bot, ← h]
-  classical exact IsGalois.of_fixed_field E (⊤ : Subgroup (E ≃ₐ[F] E))
+  classical
 #align is_galois.of_fixed_field_eq_bot IsGalois.of_fixedField_eq_bot
 -/
 
@@ -439,14 +426,6 @@ theorem of_card_aut_eq_finrank [FiniteDimensional F E]
   apply of_fixed_field_eq_bot
   have p : 0 < finrank (IntermediateField.fixedField (⊤ : Subgroup (E ≃ₐ[F] E))) E := finrank_pos
   classical
-  rw [← IntermediateField.finrank_eq_one_iff, ← mul_left_inj' (ne_of_lt p).symm,
-    finrank_mul_finrank, ← h, one_mul, IntermediateField.finrank_fixedField_eq_card]
-  apply Fintype.card_congr
-  exact
-    { toFun := fun g => ⟨g, Subgroup.mem_top g⟩
-      invFun := coe
-      left_inv := fun g => rfl
-      right_inv := fun _ => by ext; rfl }
 #align is_galois.of_card_aut_eq_finrank IsGalois.of_card_aut_eq_finrank
 -/
 

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

@@ -469,7 +469,7 @@ theorem of_separable_splitting_field_aux [hFE : FiniteDimensional F E] [sp : p.I
     Fintype.card (K⟮⟯.restrictScalars F →ₐ[F] E) = Fintype.card (K →ₐ[F] E) * finrank K K⟮⟯ :=
   by
   have h : IsIntegral K x :=
-    isIntegral_of_isScalarTower (isIntegral_of_noetherian (IsNoetherian.iff_fg.2 hFE) x)
+    IsIntegral.tower_top (isIntegral_of_noetherian (IsNoetherian.iff_fg.2 hFE) x)
   have h1 : p ≠ 0 := fun hp => by rwa [hp, Polynomial.map_zero, Polynomial.roots_zero] at hx 
   have h2 : minpoly K x ∣ p.map (algebraMap F K) :=
     by

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

@@ -3,9 +3,9 @@ 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.PrimitiveElement
-import Mathbin.FieldTheory.Fixed
-import Mathbin.GroupTheory.GroupAction.FixingSubgroup
+import FieldTheory.PrimitiveElement
+import FieldTheory.Fixed
+import GroupTheory.GroupAction.FixingSubgroup
 
 #align_import field_theory.galois from "leanprover-community/mathlib"@"2a0ce625dbb0ffbc7d1316597de0b25c1ec75303"
 

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

@@ -2,16 +2,13 @@
 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.galois
-! leanprover-community/mathlib commit 2a0ce625dbb0ffbc7d1316597de0b25c1ec75303
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.FieldTheory.PrimitiveElement
 import Mathbin.FieldTheory.Fixed
 import Mathbin.GroupTheory.GroupAction.FixingSubgroup
 
+#align_import field_theory.galois from "leanprover-community/mathlib"@"2a0ce625dbb0ffbc7d1316597de0b25c1ec75303"
+
 /-!
 # Galois Extensions
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/2a0ce625dbb0ffbc7d1316597de0b25c1ec75303

@@ -48,20 +48,24 @@ section
 
 variable (F : Type _) [Field F] (E : Type _) [Field E] [Algebra F E]
 
+#print IsGalois /-
 /-- A field extension E/F is galois if it is both separable and normal. Note that in mathlib
 a separable extension of fields is by definition algebraic. -/
 class IsGalois : Prop where
   [to_isSeparable : IsSeparable F E]
   [to_normal : Normal F E]
 #align is_galois IsGalois
+-/
 
 variable {F E}
 
+#print isGalois_iff /-
 theorem isGalois_iff : IsGalois F E ↔ IsSeparable F E ∧ Normal F E :=
   ⟨fun h => ⟨h.1, h.2⟩, fun h =>
     { to_isSeparable := h.1
       to_normal := h.2 }⟩
 #align is_galois_iff isGalois_iff
+-/
 
 attribute [instance 100] IsGalois.to_isSeparable IsGalois.to_normal
 
@@ -70,30 +74,40 @@ variable (F E)
 
 namespace IsGalois
 
+#print IsGalois.self /-
 instance self : IsGalois F F :=
   ⟨⟩
 #align is_galois.self IsGalois.self
+-/
 
 variable (F) {E}
 
+#print IsGalois.integral /-
 theorem integral [IsGalois F E] (x : E) : IsIntegral F x :=
   to_normal.IsIntegral x
 #align is_galois.integral IsGalois.integral
+-/
 
+#print IsGalois.separable /-
 theorem separable [IsGalois F E] (x : E) : (minpoly F x).Separable :=
   IsSeparable.separable F x
 #align is_galois.separable IsGalois.separable
+-/
 
+#print IsGalois.splits /-
 theorem splits [IsGalois F E] (x : E) : (minpoly F x).Splits (algebraMap F E) :=
   Normal.splits' x
 #align is_galois.splits IsGalois.splits
+-/
 
 variable (F E)
 
+#print IsGalois.of_fixed_field /-
 instance of_fixed_field (G : Type _) [Group G] [Finite G] [MulSemiringAction G E] :
     IsGalois (FixedPoints.subfield G E) E :=
   ⟨⟩
 #align is_galois.of_fixed_field IsGalois.of_fixed_field
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:192:11: unsupported (impossible) -/
@@ -102,6 +116,7 @@ instance of_fixed_field (G : Type _) [Group G] [Finite G] [MulSemiringAction G E
 /- ./././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 IsGalois.IntermediateField.AdjoinSimple.card_aut_eq_finrank /-
 theorem IntermediateField.AdjoinSimple.card_aut_eq_finrank [FiniteDimensional F E] {α : E}
     (hα : IsIntegral F α) (h_sep : (minpoly F α).Separable)
     (h_splits : (minpoly F α).Splits (algebraMap F F⟮⟯)) :
@@ -112,10 +127,12 @@ theorem IntermediateField.AdjoinSimple.card_aut_eq_finrank [FiniteDimensional F
   rw [← IntermediateField.card_algHom_adjoin_integral F hα h_sep h_splits]
   exact Fintype.card_congr (algEquivEquivAlgHom F F⟮⟯)
 #align is_galois.intermediate_field.adjoin_simple.card_aut_eq_finrank IsGalois.IntermediateField.AdjoinSimple.card_aut_eq_finrank
+-/
 
 /- ./././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 IsGalois.card_aut_eq_finrank /-
 theorem card_aut_eq_finrank [FiniteDimensional F E] [IsGalois F E] :
     Fintype.card (E ≃ₐ[F] E) = finrank F E :=
   by
@@ -143,6 +160,7 @@ theorem card_aut_eq_finrank [FiniteDimensional F E] [IsGalois F E] :
   · intro ϕ; ext1; simp only [trans_apply, apply_symm_apply]
   · intro ϕ; ext1; simp only [trans_apply, symm_apply_apply]
 #align is_galois.card_aut_eq_finrank IsGalois.card_aut_eq_finrank
+-/
 
 end IsGalois
 
@@ -154,19 +172,24 @@ variable (F K E : Type _) [Field F] [Field K] [Field E] {E' : Type _} [Field E']
 
 variable [Algebra F K] [Algebra F E] [Algebra K E] [IsScalarTower F K E]
 
+#print IsGalois.tower_top_of_isGalois /-
 theorem IsGalois.tower_top_of_isGalois [IsGalois F E] : IsGalois K E :=
   { to_isSeparable := isSeparable_tower_top_of_isSeparable F K E
     to_normal := Normal.tower_top_of_normal F K E }
 #align is_galois.tower_top_of_is_galois IsGalois.tower_top_of_isGalois
+-/
 
 variable {F E}
 
+#print IsGalois.tower_top_intermediateField /-
 -- see Note [lower instance priority]
 instance (priority := 100) IsGalois.tower_top_intermediateField (K : IntermediateField F E)
     [h : IsGalois F E] : IsGalois K E :=
   IsGalois.tower_top_of_isGalois F K E
 #align is_galois.tower_top_intermediate_field IsGalois.tower_top_intermediateField
+-/
 
+#print isGalois_iff_isGalois_bot /-
 theorem isGalois_iff_isGalois_bot : IsGalois (⊥ : IntermediateField F E) E ↔ IsGalois F E :=
   by
   constructor
@@ -174,23 +197,32 @@ theorem isGalois_iff_isGalois_bot : IsGalois (⊥ : IntermediateField F E) E ↔
     exact IsGalois.tower_top_of_isGalois (⊥ : IntermediateField F E) F E
   · intro h; infer_instance
 #align is_galois_iff_is_galois_bot isGalois_iff_isGalois_bot
+-/
 
+#print IsGalois.of_algEquiv /-
 theorem IsGalois.of_algEquiv [h : IsGalois F E] (f : E ≃ₐ[F] E') : IsGalois F E' :=
   { to_isSeparable := IsSeparable.of_algHom F E f.symm
     to_normal := Normal.of_algEquiv f }
 #align is_galois.of_alg_equiv IsGalois.of_algEquiv
+-/
 
+#print AlgEquiv.transfer_galois /-
 theorem AlgEquiv.transfer_galois (f : E ≃ₐ[F] E') : IsGalois F E ↔ IsGalois F E' :=
   ⟨fun h => IsGalois.of_algEquiv f, fun h => IsGalois.of_algEquiv f.symm⟩
 #align alg_equiv.transfer_galois AlgEquiv.transfer_galois
+-/
 
+#print isGalois_iff_isGalois_top /-
 theorem isGalois_iff_isGalois_top : IsGalois F (⊤ : IntermediateField F E) ↔ IsGalois F E :=
   (IntermediateField.topEquiv : (⊤ : IntermediateField F E) ≃ₐ[F] E).transfer_galois
 #align is_galois_iff_is_galois_top isGalois_iff_isGalois_top
+-/
 
+#print isGalois_bot /-
 instance isGalois_bot : IsGalois F (⊥ : IntermediateField F E) :=
   (IntermediateField.botEquiv F E).transfer_galois.mpr (IsGalois.self F)
 #align is_galois_bot isGalois_bot
+-/
 
 end IsGaloisTower
 
@@ -200,6 +232,7 @@ variable {F : Type _} [Field F] {E : Type _} [Field E] [Algebra F E]
 
 variable (H : Subgroup (E ≃ₐ[F] E)) (K : IntermediateField F E)
 
+#print FixedPoints.intermediateField /-
 /-- The intermediate field of fixed points fixed by a monoid action that commutes with the
 `F`-action on `E`. -/
 def FixedPoints.intermediateField (M : Type _) [Monoid M] [MulSemiringAction M E]
@@ -208,28 +241,38 @@ def FixedPoints.intermediateField (M : Type _) [Monoid M] [MulSemiringAction M E
     carrier := MulAction.fixedPoints M E
     algebraMap_mem' := fun a g => by rw [Algebra.algebraMap_eq_smul_one, smul_comm, smul_one] }
 #align fixed_points.intermediate_field FixedPoints.intermediateField
+-/
 
 namespace IntermediateField
 
+#print IntermediateField.fixedField /-
 /-- The intermediate_field fixed by a subgroup -/
 def fixedField : IntermediateField F E :=
   FixedPoints.intermediateField H
 #align intermediate_field.fixed_field IntermediateField.fixedField
+-/
 
+#print IntermediateField.finrank_fixedField_eq_card /-
 theorem finrank_fixedField_eq_card [FiniteDimensional F E] [DecidablePred (· ∈ H)] :
     finrank (fixedField H) E = Fintype.card H :=
   FixedPoints.finrank_eq_card H E
 #align intermediate_field.finrank_fixed_field_eq_card IntermediateField.finrank_fixedField_eq_card
+-/
 
+#print IntermediateField.fixingSubgroup /-
 /-- The subgroup fixing an intermediate_field -/
 def fixingSubgroup : Subgroup (E ≃ₐ[F] E) :=
   fixingSubgroup (E ≃ₐ[F] E) (K : Set E)
 #align intermediate_field.fixing_subgroup IntermediateField.fixingSubgroup
+-/
 
+#print IntermediateField.le_iff_le /-
 theorem le_iff_le : K ≤ fixedField H ↔ H ≤ fixingSubgroup K :=
   ⟨fun h g hg x => h (Subtype.mem x) ⟨g, hg⟩, fun h x hx g => h (Subtype.mem g) ⟨x, hx⟩⟩
 #align intermediate_field.le_iff_le IntermediateField.le_iff_le
+-/
 
+#print IntermediateField.fixingSubgroupEquiv /-
 /-- The fixing_subgroup of `K : intermediate_field F E` is isomorphic to `E ≃ₐ[K] E` -/
 def fixingSubgroupEquiv : fixingSubgroup K ≃* E ≃ₐ[K] E
     where
@@ -239,7 +282,9 @@ def fixingSubgroupEquiv : fixingSubgroup K ≃* E ≃ₐ[K] E
   right_inv _ := by ext; rfl
   map_mul' _ _ := by ext; rfl
 #align intermediate_field.fixing_subgroup_equiv IntermediateField.fixingSubgroupEquiv
+-/
 
+#print IntermediateField.fixingSubgroup_fixedField /-
 theorem fixingSubgroup_fixedField [FiniteDimensional F E] : fixingSubgroup (fixedField H) = H :=
   by
   have H_le : H ≤ fixingSubgroup (fixed_field H) := (le_iff_le _ _).mp le_rfl
@@ -255,7 +300,9 @@ theorem fixingSubgroup_fixedField [FiniteDimensional F E] : fixingSubgroup (fixe
   refine' (algEquivEquivAlgHom (fixed_field H) E).toEquiv.symm.trans _
   exact (fixing_subgroup_equiv (fixed_field H)).toEquiv.symm
 #align intermediate_field.fixing_subgroup_fixed_field IntermediateField.fixingSubgroup_fixedField
+-/
 
+#print IntermediateField.fixedField.algebra /-
 instance fixedField.algebra : Algebra K (fixedField (fixingSubgroup K))
     where
   smul x y :=
@@ -269,15 +316,19 @@ instance fixedField.algebra : Algebra K (fixedField (fixingSubgroup K))
   commutes' _ _ := mul_comm _ _
   smul_def' _ _ := rfl
 #align intermediate_field.fixed_field.algebra IntermediateField.fixedField.algebra
+-/
 
+#print IntermediateField.fixedField.isScalarTower /-
 instance fixedField.isScalarTower : IsScalarTower K (fixedField (fixingSubgroup K)) E :=
   ⟨fun _ _ _ => mul_assoc _ _ _⟩
 #align intermediate_field.fixed_field.is_scalar_tower IntermediateField.fixedField.isScalarTower
+-/
 
 end IntermediateField
 
 namespace IsGalois
 
+#print IsGalois.fixedField_fixingSubgroup /-
 theorem fixedField_fixingSubgroup [FiniteDimensional F E] [h : IsGalois F E] :
     IntermediateField.fixedField (IntermediateField.fixingSubgroup K) = K :=
   by
@@ -291,7 +342,9 @@ theorem fixedField_fixingSubgroup [FiniteDimensional F E] [h : IsGalois F E] :
     Fintype.card_congr (IntermediateField.fixingSubgroupEquiv K).toEquiv]
   exact (card_aut_eq_finrank K E).symm
 #align is_galois.fixed_field_fixing_subgroup IsGalois.fixedField_fixingSubgroup
+-/
 
+#print IsGalois.card_fixingSubgroup_eq_finrank /-
 theorem card_fixingSubgroup_eq_finrank [DecidablePred (· ∈ IntermediateField.fixingSubgroup K)]
     [FiniteDimensional F E] [IsGalois F E] :
     Fintype.card (IntermediateField.fixingSubgroup K) = finrank K E := by
@@ -299,7 +352,9 @@ theorem card_fixingSubgroup_eq_finrank [DecidablePred (· ∈ IntermediateField.
     rhs
     rw [← fixed_field_fixing_subgroup K, IntermediateField.finrank_fixedField_eq_card]
 #align is_galois.card_fixing_subgroup_eq_finrank IsGalois.card_fixingSubgroup_eq_finrank
+-/
 
+#print IsGalois.intermediateFieldEquivSubgroup /-
 /-- The Galois correspondence from intermediate fields to subgroups -/
 def intermediateFieldEquivSubgroup [FiniteDimensional F E] [IsGalois F E] :
     IntermediateField F E ≃o (Subgroup (E ≃ₐ[F] E))ᵒᵈ
@@ -313,7 +368,9 @@ def intermediateFieldEquivSubgroup [FiniteDimensional F E] [IsGalois F E] :
     rw [← fixed_field_fixing_subgroup L, IntermediateField.le_iff_le, fixed_field_fixing_subgroup L]
     rfl
 #align is_galois.intermediate_field_equiv_subgroup IsGalois.intermediateFieldEquivSubgroup
+-/
 
+#print IsGalois.galoisInsertionIntermediateFieldSubgroup /-
 /-- The Galois correspondence as a galois_insertion -/
 def galoisInsertionIntermediateFieldSubgroup [FiniteDimensional F E] :
     GaloisInsertion
@@ -327,7 +384,9 @@ def galoisInsertionIntermediateFieldSubgroup [FiniteDimensional F E] :
   le_l_u H := le_of_eq (IntermediateField.fixingSubgroup_fixedField H).symm
   choice_eq K _ := rfl
 #align is_galois.galois_insertion_intermediate_field_subgroup IsGalois.galoisInsertionIntermediateFieldSubgroup
+-/
 
+#print IsGalois.galoisCoinsertionIntermediateFieldSubgroup /-
 /-- The Galois correspondence as a galois_coinsertion -/
 def galoisCoinsertionIntermediateFieldSubgroup [FiniteDimensional F E] [IsGalois F E] :
     GaloisCoinsertion
@@ -341,6 +400,7 @@ def galoisCoinsertionIntermediateFieldSubgroup [FiniteDimensional F E] [IsGalois
   u_l_le K := le_of_eq (fixedField_fixingSubgroup K)
   choice_eq H _ := rfl
 #align is_galois.galois_coinsertion_intermediate_field_subgroup IsGalois.galoisCoinsertionIntermediateFieldSubgroup
+-/
 
 end IsGalois
 
@@ -352,6 +412,7 @@ variable (F : Type _) [Field F] (E : Type _) [Field E] [Algebra F E]
 
 namespace IsGalois
 
+#print IsGalois.is_separable_splitting_field /-
 theorem is_separable_splitting_field [FiniteDimensional F E] [IsGalois F E] :
     ∃ p : F[X], p.Separable ∧ p.IsSplittingField F E :=
   by
@@ -363,14 +424,18 @@ theorem is_separable_splitting_field [FiniteDimensional F E] [IsGalois F E] :
   rw [Set.singleton_subset_iff, Polynomial.mem_rootSet]
   exact ⟨minpoly.ne_zero (integral F α), minpoly.aeval _ _⟩
 #align is_galois.is_separable_splitting_field IsGalois.is_separable_splitting_field
+-/
 
+#print IsGalois.of_fixedField_eq_bot /-
 theorem of_fixedField_eq_bot [FiniteDimensional F E]
     (h : IntermediateField.fixedField (⊤ : Subgroup (E ≃ₐ[F] E)) = ⊥) : IsGalois F E :=
   by
   rw [← isGalois_iff_isGalois_bot, ← h]
   classical exact IsGalois.of_fixed_field E (⊤ : Subgroup (E ≃ₐ[F] E))
 #align is_galois.of_fixed_field_eq_bot IsGalois.of_fixedField_eq_bot
+-/
 
+#print IsGalois.of_card_aut_eq_finrank /-
 theorem of_card_aut_eq_finrank [FiniteDimensional F E]
     (h : Fintype.card (E ≃ₐ[F] E) = finrank F E) : IsGalois F E :=
   by
@@ -386,6 +451,7 @@ theorem of_card_aut_eq_finrank [FiniteDimensional F E]
       left_inv := fun g => rfl
       right_inv := fun _ => by ext; rfl }
 #align is_galois.of_card_aut_eq_finrank IsGalois.of_card_aut_eq_finrank
+-/
 
 variable {F} {E} {p : F[X]}
 
@@ -396,6 +462,7 @@ variable {F} {E} {p : F[X]}
 /- ./././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 IsGalois.of_separable_splitting_field_aux /-
 theorem of_separable_splitting_field_aux [hFE : FiniteDimensional F E] [sp : p.IsSplittingField F E]
     (hp : p.Separable) (K : Type _) [Field K] [Algebra F K] [Algebra K E] [IsScalarTower F K E]
     {x : E} (hx : x ∈ (p.map (algebraMap F E)).roots)
@@ -433,7 +500,9 @@ theorem of_separable_splitting_field_aux [hFE : FiniteDimensional F E] [sp : p.I
     rw [Polynomial.splits_map_iff, ← IsScalarTower.algebraMap_eq]
     exact sp.splits
 #align is_galois.of_separable_splitting_field_aux IsGalois.of_separable_splitting_field_aux
+-/
 
+#print IsGalois.of_separable_splitting_field /-
 theorem of_separable_splitting_field [sp : p.IsSplittingField F E] (hp : p.Separable) :
     IsGalois F E :=
   by
@@ -472,9 +541,11 @@ theorem of_separable_splitting_field [sp : p.IsSplittingField F E] (hp : p.Separ
   refine' LinearEquiv.finrank_eq _
   rfl
 #align is_galois.of_separable_splitting_field IsGalois.of_separable_splitting_field
+-/
 
+#print IsGalois.tfae /-
 /-- Equivalent characterizations of a Galois extension of finite degree-/
-theorem tFAE [FiniteDimensional F E] :
+theorem tfae [FiniteDimensional F E] :
     TFAE
       [IsGalois F E, IntermediateField.fixedField (⊤ : Subgroup (E ≃ₐ[F] E)) = ⊥,
         Fintype.card (E ≃ₐ[F] E) = finrank F E, ∃ p : F[X], p.Separable ∧ p.IsSplittingField F E] :=
@@ -492,7 +563,8 @@ theorem tFAE [FiniteDimensional F E] :
   tfae_have 4 → 1
   · rintro ⟨h, hp1, _⟩; exact of_separable_splitting_field hp1
   tfae_finish
-#align is_galois.tfae IsGalois.tFAE
+#align is_galois.tfae IsGalois.tfae
+-/
 
 end IsGalois
 
@@ -503,17 +575,21 @@ section normalClosure
 variable (k K F : Type _) [Field k] [Field K] [Field F] [Algebra k K] [Algebra k F] [Algebra K F]
   [IsScalarTower k K F] [IsGalois k F]
 
+#print IsGalois.normalClosure /-
 instance IsGalois.normalClosure : IsGalois k (normalClosure k K F)
     where to_isSeparable := isSeparable_tower_bot_of_isSeparable k _ F
 #align is_galois.normal_closure IsGalois.normalClosure
+-/
 
 end normalClosure
 
 section IsAlgClosure
 
+#print IsAlgClosure.isGalois /-
 instance (priority := 100) IsAlgClosure.isGalois (k K : Type _) [Field k] [Field K] [Algebra k K]
     [IsAlgClosure k K] [CharZero k] : IsGalois k K where
 #align is_alg_closure.is_galois IsAlgClosure.isGalois
+-/
 
 end IsAlgClosure
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/2a0ce625dbb0ffbc7d1316597de0b25c1ec75303

@@ -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.galois
-! leanprover-community/mathlib commit 9fb8964792b4237dac6200193a0d533f1b3f7423
+! leanprover-community/mathlib commit 2a0ce625dbb0ffbc7d1316597de0b25c1ec75303
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -15,6 +15,9 @@ import Mathbin.GroupTheory.GroupAction.FixingSubgroup
 /-!
 # Galois Extensions
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 In this file we define Galois extensions as extensions which are both separable and normal.
 
 ## Main definitions

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

@@ -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.galois
-! leanprover-community/mathlib commit 00f91228655eecdcd3ac97a7fd8dbcb139fe990a
+! leanprover-community/mathlib commit 9fb8964792b4237dac6200193a0d533f1b3f7423
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -357,11 +357,8 @@ theorem is_separable_splitting_field [FiniteDimensional F E] [IsGalois F E] :
   rw [eq_top_iff, ← IntermediateField.top_toSubalgebra, ← h1]
   rw [IntermediateField.adjoin_simple_toSubalgebra_of_integral (integral F α)]
   apply Algebra.adjoin_mono
-  rw [Set.singleton_subset_iff, Finset.mem_coe, Multiset.mem_toFinset, Polynomial.mem_roots]
-  · dsimp only [Polynomial.IsRoot]
-    rw [Polynomial.eval_map, ← Polynomial.aeval_def]
-    exact minpoly.aeval _ _
-  · exact Polynomial.map_ne_zero (minpoly.ne_zero (integral F α))
+  rw [Set.singleton_subset_iff, Polynomial.mem_rootSet]
+  exact ⟨minpoly.ne_zero (integral F α), minpoly.aeval _ _⟩
 #align is_galois.is_separable_splitting_field IsGalois.is_separable_splitting_field
 
 theorem of_fixedField_eq_bot [FiniteDimensional F E]
@@ -443,7 +440,7 @@ theorem of_separable_splitting_field [sp : p.IsSplittingField F E] (hp : p.Separ
   have adjoin_root : IntermediateField.adjoin F ↑s = ⊤ :=
     by
     apply IntermediateField.toSubalgebra_injective
-    rw [IntermediateField.top_toSubalgebra, ← top_le_iff, ← sp.adjoin_roots]
+    rw [IntermediateField.top_toSubalgebra, ← top_le_iff, ← sp.adjoin_root_set]
     apply IntermediateField.algebra_adjoin_le_adjoin
   let P : IntermediateField F E → Prop := fun K => Fintype.card (K →ₐ[F] E) = finrank F K
   suffices P (IntermediateField.adjoin F ↑s)

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

@@ -241,16 +241,16 @@ theorem fixingSubgroup_fixedField [FiniteDimensional F E] : fixingSubgroup (fixe
   by
   have H_le : H ≤ fixingSubgroup (fixed_field H) := (le_iff_le _ _).mp le_rfl
   classical
-    suffices Fintype.card H = Fintype.card (fixingSubgroup (fixed_field H)) by
-      exact
-        SetLike.coe_injective
-          (Set.eq_of_inclusion_surjective
-              ((Fintype.bijective_iff_injective_and_card (Set.inclusion H_le)).mpr
-                  ⟨Set.inclusion_injective H_le, this⟩).2).symm
-    apply Fintype.card_congr
-    refine' (FixedPoints.toAlgHomEquiv H E).trans _
-    refine' (algEquivEquivAlgHom (fixed_field H) E).toEquiv.symm.trans _
-    exact (fixing_subgroup_equiv (fixed_field H)).toEquiv.symm
+  suffices Fintype.card H = Fintype.card (fixingSubgroup (fixed_field H)) by
+    exact
+      SetLike.coe_injective
+        (Set.eq_of_inclusion_surjective
+            ((Fintype.bijective_iff_injective_and_card (Set.inclusion H_le)).mpr
+                ⟨Set.inclusion_injective H_le, this⟩).2).symm
+  apply Fintype.card_congr
+  refine' (FixedPoints.toAlgHomEquiv H E).trans _
+  refine' (algEquivEquivAlgHom (fixed_field H) E).toEquiv.symm.trans _
+  exact (fixing_subgroup_equiv (fixed_field H)).toEquiv.symm
 #align intermediate_field.fixing_subgroup_fixed_field IntermediateField.fixingSubgroup_fixedField
 
 instance fixedField.algebra : Algebra K (fixedField (fixingSubgroup K))
@@ -284,9 +284,9 @@ theorem fixedField_fixingSubgroup [FiniteDimensional F E] [h : IsGalois F E] :
     finrank K E = finrank (IntermediateField.fixedField (IntermediateField.fixingSubgroup K)) E by
     exact (IntermediateField.eq_of_le_of_finrank_eq' K_le this).symm
   classical
-    rw [IntermediateField.finrank_fixedField_eq_card,
-      Fintype.card_congr (IntermediateField.fixingSubgroupEquiv K).toEquiv]
-    exact (card_aut_eq_finrank K E).symm
+  rw [IntermediateField.finrank_fixedField_eq_card,
+    Fintype.card_congr (IntermediateField.fixingSubgroupEquiv K).toEquiv]
+  exact (card_aut_eq_finrank K E).symm
 #align is_galois.fixed_field_fixing_subgroup IsGalois.fixedField_fixingSubgroup
 
 theorem card_fixingSubgroup_eq_finrank [DecidablePred (· ∈ IntermediateField.fixingSubgroup K)]
@@ -377,14 +377,14 @@ theorem of_card_aut_eq_finrank [FiniteDimensional F E]
   apply of_fixed_field_eq_bot
   have p : 0 < finrank (IntermediateField.fixedField (⊤ : Subgroup (E ≃ₐ[F] E))) E := finrank_pos
   classical
-    rw [← IntermediateField.finrank_eq_one_iff, ← mul_left_inj' (ne_of_lt p).symm,
-      finrank_mul_finrank, ← h, one_mul, IntermediateField.finrank_fixedField_eq_card]
-    apply Fintype.card_congr
-    exact
-      { toFun := fun g => ⟨g, Subgroup.mem_top g⟩
-        invFun := coe
-        left_inv := fun g => rfl
-        right_inv := fun _ => by ext; rfl }
+  rw [← IntermediateField.finrank_eq_one_iff, ← mul_left_inj' (ne_of_lt p).symm,
+    finrank_mul_finrank, ← h, one_mul, IntermediateField.finrank_fixedField_eq_card]
+  apply Fintype.card_congr
+  exact
+    { toFun := fun g => ⟨g, Subgroup.mem_top g⟩
+      invFun := coe
+      left_inv := fun g => rfl
+      right_inv := fun _ => by ext; rfl }
 #align is_galois.of_card_aut_eq_finrank IsGalois.of_card_aut_eq_finrank
 
 variable {F} {E} {p : F[X]}

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

@@ -406,7 +406,7 @@ theorem of_separable_splitting_field_aux [hFE : FiniteDimensional F E] [sp : p.I
   by
   have h : IsIntegral K x :=
     isIntegral_of_isScalarTower (isIntegral_of_noetherian (IsNoetherian.iff_fg.2 hFE) x)
-  have h1 : p ≠ 0 := fun hp => by rwa [hp, Polynomial.map_zero, Polynomial.roots_zero] at hx
+  have h1 : p ≠ 0 := fun hp => by rwa [hp, Polynomial.map_zero, Polynomial.roots_zero] at hx 
   have h2 : minpoly K x ∣ p.map (algebraMap F K) :=
     by
     apply minpoly.dvd
@@ -415,9 +415,9 @@ theorem of_separable_splitting_field_aux [hFE : FiniteDimensional F E] [sp : p.I
     exact (Polynomial.mem_roots (Polynomial.map_ne_zero h1)).mp hx
   let key_equiv :
     (K⟮⟯.restrictScalars F →ₐ[F] E) ≃
-      Σf : K →ₐ[F] E, @AlgHom K K⟮⟯ E _ _ _ _ (RingHom.toAlgebra f) :=
+      Σ f : K →ₐ[F] E, @AlgHom K K⟮⟯ E _ _ _ _ (RingHom.toAlgebra f) :=
     by
-    change (K⟮⟯ →ₐ[F] E) ≃ Σf : K →ₐ[F] E, _
+    change (K⟮⟯ →ₐ[F] E) ≃ Σ f : K →ₐ[F] E, _
     exact algHomEquivSigma
   haveI : ∀ f : K →ₐ[F] E, Fintype (@AlgHom K K⟮⟯ E _ _ _ _ (RingHom.toAlgebra f)) := fun f =>
     by
@@ -448,7 +448,7 @@ theorem of_separable_splitting_field [sp : p.IsSplittingField F E] (hp : p.Separ
   let P : IntermediateField F E → Prop := fun K => Fintype.card (K →ₐ[F] E) = finrank F K
   suffices P (IntermediateField.adjoin F ↑s)
     by
-    rw [AdjoinRoot] at this
+    rw [AdjoinRoot] at this 
     apply of_card_aut_eq_finrank
     rw [← Eq.trans this (LinearEquiv.finrank_eq intermediate_field.top_equiv.to_linear_equiv)]
     exact
@@ -459,9 +459,9 @@ theorem of_separable_splitting_field [sp : p.IsSplittingField F E] (hp : p.Separ
   · have key :=
       IntermediateField.card_algHom_adjoin_integral F
         (show IsIntegral F (0 : E) from isIntegral_zero)
-    rw [minpoly.zero, Polynomial.natDegree_X] at key
+    rw [minpoly.zero, Polynomial.natDegree_X] at key 
     specialize key Polynomial.separable_X (Polynomial.splits_X (algebraMap F E))
-    rw [← @Subalgebra.finrank_bot F E _ _ _, ← IntermediateField.bot_toSubalgebra] at key
+    rw [← @Subalgebra.finrank_bot F E _ _ _, ← IntermediateField.bot_toSubalgebra] at key 
     refine' Eq.trans _ key
     apply Fintype.card_congr
     rw [IntermediateField.adjoin_zero]
@@ -482,9 +482,9 @@ theorem tFAE [FiniteDimensional F E] :
   tfae_have 1 → 2
   · exact fun h => OrderIso.map_bot (@intermediate_field_equiv_subgroup F _ E _ _ _ h).symm
   tfae_have 1 → 3
-  · intro ; exact card_aut_eq_finrank F E
+  · intro; exact card_aut_eq_finrank F E
   tfae_have 1 → 4
-  · intro ; exact is_separable_splitting_field F E
+  · intro; exact is_separable_splitting_field F E
   tfae_have 2 → 1
   · exact of_fixed_field_eq_bot F E
   tfae_have 3 → 1

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

@@ -37,7 +37,7 @@ Together, these two results prove the Galois correspondence.
 -/
 
 
-open Polynomial
+open scoped Polynomial
 
 open FiniteDimensional AlgEquiv
 

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

@@ -119,13 +119,8 @@ theorem card_aut_eq_finrank [FiniteDimensional F E] [IsGalois F E] :
   cases' Field.exists_primitive_element F E with α hα
   let iso : F⟮⟯ ≃ₐ[F] E :=
     { toFun := fun e => e.val
-      invFun := fun e =>
-        ⟨e, by
-          rw [hα]
-          exact IntermediateField.mem_top⟩
-      left_inv := fun _ => by
-        ext
-        rfl
+      invFun := fun e => ⟨e, by rw [hα]; exact IntermediateField.mem_top⟩
+      left_inv := fun _ => by ext; rfl
       right_inv := fun _ => rfl
       map_mul' := fun _ _ => rfl
       map_add' := fun _ _ => rfl
@@ -134,9 +129,7 @@ theorem card_aut_eq_finrank [FiniteDimensional F E] [IsGalois F E] :
   have h_sep : (minpoly F α).Separable := IsGalois.separable F α
   have h_splits : (minpoly F α).Splits (algebraMap F E) := IsGalois.splits F α
   replace h_splits : Polynomial.Splits (algebraMap F F⟮⟯) (minpoly F α)
-  · have p : iso.symm.to_alg_hom.to_ring_hom.comp (algebraMap F E) = algebraMap F ↥F⟮⟯ :=
-      by
-      ext
+  · have p : iso.symm.to_alg_hom.to_ring_hom.comp (algebraMap F E) = algebraMap F ↥F⟮⟯ := by ext;
       simp
     simpa [p] using
       Polynomial.splits_comp_of_splits (algebraMap F E) iso.symm.to_alg_hom.to_ring_hom h_splits
@@ -144,12 +137,8 @@ theorem card_aut_eq_finrank [FiniteDimensional F E] [IsGalois F E] :
   rw [← intermediate_field.adjoin_simple.card_aut_eq_finrank F E H h_sep h_splits]
   apply Fintype.card_congr
   apply Equiv.mk (fun ϕ => iso.trans (trans ϕ iso.symm)) fun ϕ => iso.symm.trans (trans ϕ iso)
-  · intro ϕ
-    ext1
-    simp only [trans_apply, apply_symm_apply]
-  · intro ϕ
-    ext1
-    simp only [trans_apply, symm_apply_apply]
+  · intro ϕ; ext1; simp only [trans_apply, apply_symm_apply]
+  · intro ϕ; ext1; simp only [trans_apply, symm_apply_apply]
 #align is_galois.card_aut_eq_finrank IsGalois.card_aut_eq_finrank
 
 end IsGalois
@@ -180,8 +169,7 @@ theorem isGalois_iff_isGalois_bot : IsGalois (⊥ : IntermediateField F E) E ↔
   constructor
   · intro h
     exact IsGalois.tower_top_of_isGalois (⊥ : IntermediateField F E) F E
-  · intro h
-    infer_instance
+  · intro h; infer_instance
 #align is_galois_iff_is_galois_bot isGalois_iff_isGalois_bot
 
 theorem IsGalois.of_algEquiv [h : IsGalois F E] (f : E ≃ₐ[F] E') : IsGalois F E' :=
@@ -244,15 +232,9 @@ def fixingSubgroupEquiv : fixingSubgroup K ≃* E ≃ₐ[K] E
     where
   toFun ϕ := { AlgEquiv.toRingEquiv ↑ϕ with commutes' := ϕ.Mem }
   invFun ϕ := ⟨ϕ.restrictScalars _, ϕ.commutes⟩
-  left_inv _ := by
-    ext
-    rfl
-  right_inv _ := by
-    ext
-    rfl
-  map_mul' _ _ := by
-    ext
-    rfl
+  left_inv _ := by ext; rfl
+  right_inv _ := by ext; rfl
+  map_mul' _ _ := by ext; rfl
 #align intermediate_field.fixing_subgroup_equiv IntermediateField.fixingSubgroupEquiv
 
 theorem fixingSubgroup_fixedField [FiniteDimensional F E] : fixingSubgroup (fixedField H) = H :=
@@ -402,9 +384,7 @@ theorem of_card_aut_eq_finrank [FiniteDimensional F E]
       { toFun := fun g => ⟨g, Subgroup.mem_top g⟩
         invFun := coe
         left_inv := fun g => rfl
-        right_inv := fun _ => by
-          ext
-          rfl }
+        right_inv := fun _ => by ext; rfl }
 #align is_galois.of_card_aut_eq_finrank IsGalois.of_card_aut_eq_finrank
 
 variable {F} {E} {p : F[X]}
@@ -447,8 +427,7 @@ theorem of_separable_splitting_field_aux [hFE : FiniteDimensional F E] [sp : p.I
   apply Finset.sum_const_nat
   intro f hf
   rw [← @IntermediateField.card_algHom_adjoin_integral K _ E _ _ x E _ (RingHom.toAlgebra f) h]
-  · apply Fintype.card_congr
-    rfl
+  · apply Fintype.card_congr; rfl
   · exact Polynomial.Separable.of_dvd ((Polynomial.separable_map (algebraMap F K)).mpr hp) h2
   · refine' Polynomial.splits_of_splits_of_dvd _ (Polynomial.map_ne_zero h1) _ h2
     rw [Polynomial.splits_map_iff, ← IsScalarTower.algebraMap_eq]
@@ -503,18 +482,15 @@ theorem tFAE [FiniteDimensional F E] :
   tfae_have 1 → 2
   · exact fun h => OrderIso.map_bot (@intermediate_field_equiv_subgroup F _ E _ _ _ h).symm
   tfae_have 1 → 3
-  · intro
-    exact card_aut_eq_finrank F E
+  · intro ; exact card_aut_eq_finrank F E
   tfae_have 1 → 4
-  · intro
-    exact is_separable_splitting_field F E
+  · intro ; exact is_separable_splitting_field F E
   tfae_have 2 → 1
   · exact of_fixed_field_eq_bot F E
   tfae_have 3 → 1
   · exact of_card_aut_eq_finrank F E
   tfae_have 4 → 1
-  · rintro ⟨h, hp1, _⟩
-    exact of_separable_splitting_field hp1
+  · rintro ⟨h, hp1, _⟩; exact of_separable_splitting_field hp1
   tfae_finish
 #align is_galois.tfae IsGalois.tFAE
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/75e7fca56381d056096ce5d05e938f63a6567828

@@ -481,7 +481,7 @@ theorem of_separable_splitting_field [sp : p.IsSplittingField F E] (hp : p.Separ
       IntermediateField.card_algHom_adjoin_integral F
         (show IsIntegral F (0 : E) from isIntegral_zero)
     rw [minpoly.zero, Polynomial.natDegree_X] at key
-    specialize key Polynomial.separable_x (Polynomial.splits_X (algebraMap F E))
+    specialize key Polynomial.separable_X (Polynomial.splits_X (algebraMap F E))
     rw [← @Subalgebra.finrank_bot F E _ _ _, ← IntermediateField.bot_toSubalgebra] at key
     refine' Eq.trans _ key
     apply Fintype.card_congr

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: Thomas Browning, Patrick Lutz
 
 ! This file was ported from Lean 3 source module field_theory.galois
-! leanprover-community/mathlib commit f0c8bf9245297a541f468be517f1bde6195105e9
+! leanprover-community/mathlib commit 00f91228655eecdcd3ac97a7fd8dbcb139fe990a
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -522,6 +522,17 @@ end IsGalois
 
 end GaloisEquivalentDefinitions
 
+section normalClosure
+
+variable (k K F : Type _) [Field k] [Field K] [Field F] [Algebra k K] [Algebra k F] [Algebra K F]
+  [IsScalarTower k K F] [IsGalois k F]
+
+instance IsGalois.normalClosure : IsGalois k (normalClosure k K F)
+    where to_isSeparable := isSeparable_tower_bot_of_isSeparable k _ F
+#align is_galois.normal_closure IsGalois.normalClosure
+
+end normalClosure
+
 section IsAlgClosure
 
 instance (priority := 100) IsAlgClosure.isGalois (k K : Type _) [Field k] [Field K] [Algebra k K]

mathlib commit https://github.com/leanprover-community/mathlib/commit/738054fa93d43512da144ec45ce799d18fd44248

@@ -4,11 +4,10 @@ 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.galois
-! leanprover-community/mathlib commit 0ac3057eb6231d2c8dfcd46767cf4a166961c0f1
+! leanprover-community/mathlib commit f0c8bf9245297a541f468be517f1bde6195105e9
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
-import Mathbin.FieldTheory.IsAlgClosed.AlgebraicClosure
 import Mathbin.FieldTheory.PrimitiveElement
 import Mathbin.FieldTheory.Fixed
 import Mathbin.GroupTheory.GroupAction.FixingSubgroup

mathlib commit https://github.com/leanprover-community/mathlib/commit/09079525fd01b3dda35e96adaa08d2f943e1648c

@@ -61,7 +61,7 @@ theorem isGalois_iff : IsGalois F E ↔ IsSeparable F E ∧ Normal F E :=
       to_normal := h.2 }⟩
 #align is_galois_iff isGalois_iff
 
-attribute [instance] IsGalois.to_isSeparable IsGalois.to_normal
+attribute [instance 100] IsGalois.to_isSeparable IsGalois.to_normal
 
 -- see Note [lower instance priority]
 variable (F E)

mathlib commit https://github.com/leanprover-community/mathlib/commit/06a655b5fcfbda03502f9158bbf6c0f1400886f9

@@ -4,11 +4,11 @@ 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.galois
-! leanprover-community/mathlib commit 70fd9563a21e7b963887c9360bd29b2393e6225a
+! leanprover-community/mathlib commit 0ac3057eb6231d2c8dfcd46767cf4a166961c0f1
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
-import Mathbin.FieldTheory.Normal
+import Mathbin.FieldTheory.IsAlgClosed.AlgebraicClosure
 import Mathbin.FieldTheory.PrimitiveElement
 import Mathbin.FieldTheory.Fixed
 import Mathbin.GroupTheory.GroupAction.FixingSubgroup
@@ -523,3 +523,11 @@ end IsGalois
 
 end GaloisEquivalentDefinitions
 
+section IsAlgClosure
+
+instance (priority := 100) IsAlgClosure.isGalois (k K : Type _) [Field k] [Field K] [Algebra k K]
+    [IsAlgClosure k K] [CharZero k] : IsGalois k K where
+#align is_alg_closure.is_galois IsAlgClosure.isGalois
+
+end IsAlgClosure
+

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce86f4e05e9a9b8da5e316b22c76ce76440c56a1

@@ -482,7 +482,7 @@ theorem of_separable_splitting_field [sp : p.IsSplittingField F E] (hp : p.Separ
       IntermediateField.card_algHom_adjoin_integral F
         (show IsIntegral F (0 : E) from isIntegral_zero)
     rw [minpoly.zero, Polynomial.natDegree_X] at key
-    specialize key Polynomial.separable_x (Polynomial.splits_x (algebraMap F E))
+    specialize key Polynomial.separable_x (Polynomial.splits_X (algebraMap F E))
     rw [← @Subalgebra.finrank_bot F E _ _ _, ← IntermediateField.bot_toSubalgebra] at key
     refine' Eq.trans _ key
     apply Fintype.card_congr

mathlib commit https://github.com/leanprover-community/mathlib/commit/38f16f960f5006c6c0c2bac7b0aba5273188f4e5

@@ -481,7 +481,7 @@ theorem of_separable_splitting_field [sp : p.IsSplittingField F E] (hp : p.Separ
   · have key :=
       IntermediateField.card_algHom_adjoin_integral F
         (show IsIntegral F (0 : E) from isIntegral_zero)
-    rw [minpoly.zero, Polynomial.natDegree_x] at key
+    rw [minpoly.zero, Polynomial.natDegree_X] at key
     specialize key Polynomial.separable_x (Polynomial.splits_x (algebraMap F E))
     rw [← @Subalgebra.finrank_bot F E _ _ _, ← IntermediateField.bot_toSubalgebra] at key
     refine' Eq.trans _ key

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

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)

@@ -132,7 +132,6 @@ end
 section IsGaloisTower
 
 variable (F K E : Type*) [Field F] [Field K] [Field E] {E' : Type*} [Field E'] [Algebra F E']
-
 variable [Algebra F K] [Algebra F E] [Algebra K E] [IsScalarTower F K E]
 
 theorem IsGalois.tower_top_of_isGalois [IsGalois F E] : IsGalois K E :=
@@ -177,7 +176,6 @@ end IsGaloisTower
 section GaloisCorrespondence
 
 variable {F : Type*} [Field F] {E : Type*} [Field E] [Algebra F E]
-
 variable (H : Subgroup (E ≃ₐ[F] E)) (K : IntermediateField F E)
 
 /-- The intermediate field of fixed points fixed by a monoid action that commutes with the
@@ -353,7 +351,6 @@ theorem of_card_aut_eq_finrank [FiniteDimensional F E]
 #align is_galois.of_card_aut_eq_finrank IsGalois.of_card_aut_eq_finrank
 
 variable {F} {E}
-
 variable {p : F[X]}
 
 theorem of_separable_splitting_field_aux [hFE : FiniteDimensional F E] [sp : p.IsSplittingField F E]

Uses smul_algebraMap to simplify the proof of FixedPoints.intermediateField.algebraMap_mem'.

Amusingly, the current tactic-mode proof is very nearly identical to the body of smul_algebraMap: https://github.com/leanprover-community/mathlib4/blob/bb9eaa6b041bc19ca8615a24fa48e463c672c150/Mathlib/Algebra/Algebra/Basic.lean#L403-L405

After making this simplificiation, I observe the time reported by trace.profiler to drop from 0.13 to 0.12 seconds.

@@ -186,7 +186,7 @@ def FixedPoints.intermediateField (M : Type*) [Monoid M] [MulSemiringAction M E]
     [SMulCommClass M F E] : IntermediateField F E :=
   { FixedPoints.subfield M E with
     carrier := MulAction.fixedPoints M E
-    algebraMap_mem' := fun a g => by rw [Algebra.algebraMap_eq_smul_one, smul_comm, smul_one] }
+    algebraMap_mem' := fun a g => smul_algebraMap g a }
 #align fixed_points.intermediate_field FixedPoints.intermediateField
 
 namespace IntermediateField

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -418,7 +418,7 @@ theorem of_separable_splitting_field [sp : p.IsSplittingField F E] (hp : p.Separ
     apply @Fintype.card_congr _ _ _ (_) _
     rw [IntermediateField.adjoin_zero]
   intro K x hx hK
-  simp only at *
+  simp only [P] at *
   -- Porting note: need to specify two implicit arguments of `finrank_mul_finrank`
   letI := K⟮x⟯.module
   letI := K⟮x⟯.isScalarTower (R := F)

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>

@@ -114,8 +114,8 @@ theorem card_aut_eq_finrank [FiniteDimensional F E] [IsGalois F E] :
   have H : IsIntegral F α := IsGalois.integral F α
   have h_sep : (minpoly F α).Separable := IsGalois.separable F α
   have h_splits : (minpoly F α).Splits (algebraMap F E) := IsGalois.splits F α
-  replace h_splits : Polynomial.Splits (algebraMap F F⟮α⟯) (minpoly F α)
-  · simpa using
+  replace h_splits : Polynomial.Splits (algebraMap F F⟮α⟯) (minpoly F α) := by
+    simpa using
       Polynomial.splits_comp_of_splits (algebraMap F E) iso.symm.toAlgHom.toRingHom h_splits
   rw [← LinearEquiv.finrank_eq iso.toLinearEquiv]
   rw [← IntermediateField.AdjoinSimple.card_aut_eq_finrank F E H h_sep h_splits]

Generalize the conditions of the tower law FiniteDimensional.finrank_mul_finrank' in FieldTheory/Tower from [CommRing F] [Algebra F K] to [Ring F] [Module F K], and remove the [Module.Finite F K] and [Module.Finite K A] conditions.

The generalized version applies to situations when we have a tower C/B/A where the A-module structure on C is induced from the B-module structure via a RingHom from A to B, and the A-module structure on B is induced by the same RingHom. In particular, it applies when the A-module structure on B and the B-module structure on C come from two RingHoms, and the A-module structure on C comes from the composition of them, regardless of whether A and B are commutative or not.

As prerequisites, I also generalized lemmas originally introduced by @kckennylau in [mathlib3#3355](https://github.com/leanprover-community/mathlib/pull/3355/files) to prove the tower law. They were split into three PRs:

• LinearAlgebra/Span #9380: add span_eq_closure and closure_induction which say that Submodule.span R s is generated by R • s as an AddSubmonoid. I feel that the existing span_induction should be replaced by closure_induction as the latter is stronger, and allow us to remove the commutativity condition in span_smul_of_span_eq_top in Algebra/Tower.

• Algebra/Tower #9382: switching from CommSemiring/Algebra to Semiring/Module here requires proving the curious lemma IsScalarTower.isLinearMap which states that for a tower of modules A/S/R, any S-linear map from S to A is also R-linear. If the map is injective, we can deduce that S/S/R also form a tower. (By ringHomEquivModuleIsScalarTower in #9381, there is therefore a canonical RingHom from R to S.)

• Lemmas for free modules over rings including finrank_mul_finrank' are moved from FieldTheory/Tower to LinearAlgebra/Dimension/Free

Zulip

Co-authored-by: Junyan Xu <junyanxu.math@gmail.com> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -420,8 +420,9 @@ theorem of_separable_splitting_field [sp : p.IsSplittingField F E] (hp : p.Separ
   intro K x hx hK
   simp only at *
   -- Porting note: need to specify two implicit arguments of `finrank_mul_finrank`
-  rw [of_separable_splitting_field_aux hp K (Multiset.mem_toFinset.mp hx), hK,
-    @finrank_mul_finrank _ _ _ _ _ _ _ K⟮x⟯.module _ K⟮x⟯.isScalarTower _]
+  letI := K⟮x⟯.module
+  letI := K⟮x⟯.isScalarTower (R := F)
+  rw [of_separable_splitting_field_aux hp K (Multiset.mem_toFinset.mp hx), hK, finrank_mul_finrank]
   symm
   refine' LinearEquiv.finrank_eq _
   rfl

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>

@@ -363,8 +363,7 @@ theorem of_separable_splitting_field_aux [hFE : FiniteDimensional F E] [sp : p.I
     [Fintype (K →ₐ[F] E)]
     [Fintype (K⟮x⟯.restrictScalars F →ₐ[F] E)] :
     Fintype.card (K⟮x⟯.restrictScalars F →ₐ[F] E) = Fintype.card (K →ₐ[F] E) * finrank K K⟮x⟯ := by
-  have h : IsIntegral K x :=
-    isIntegral_of_isScalarTower (isIntegral_of_noetherian (IsNoetherian.iff_fg.2 hFE) x)
+  have h : IsIntegral K x := (isIntegral_of_noetherian (IsNoetherian.iff_fg.2 hFE) x).tower_top
   have h1 : p ≠ 0 := fun hp => by
     rw [hp, Polynomial.aroots_zero] at hx
     exact Multiset.not_mem_zero x hx

@@ -35,7 +35,7 @@ Together, these two results prove the Galois correspondence.
 -/
 
 
-open scoped Polynomial
+open scoped Polynomial IntermediateField
 
 open FiniteDimensional AlgEquiv
 

@@ -456,8 +456,8 @@ section normalClosure
 variable (k K F : Type*) [Field k] [Field K] [Field F] [Algebra k K] [Algebra k F] [Algebra K F]
   [IsScalarTower k K F] [IsGalois k F]
 
-instance IsGalois.normalClosure : IsGalois k (normalClosure k K F)
-    where to_isSeparable := isSeparable_tower_bot_of_isSeparable k _ F
+instance IsGalois.normalClosure : IsGalois k (normalClosure k K F) where
+  to_isSeparable := isSeparable_tower_bot_of_isSeparable k _ F
 #align is_galois.normal_closure IsGalois.normalClosure
 
 end normalClosure

Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com>

@@ -358,7 +358,7 @@ variable {p : F[X]}
 
 theorem of_separable_splitting_field_aux [hFE : FiniteDimensional F E] [sp : p.IsSplittingField F E]
     (hp : p.Separable) (K : Type*) [Field K] [Algebra F K] [Algebra K E] [IsScalarTower F K E]
-    {x : E} (hx : x ∈ (p.map (algebraMap F E)).roots)
+    {x : E} (hx : x ∈ p.aroots E)
     -- these are both implied by `hFE`, but as they carry data this makes the lemma more general
     [Fintype (K →ₐ[F] E)]
     [Fintype (K⟮x⟯.restrictScalars F →ₐ[F] E)] :
@@ -366,7 +366,7 @@ theorem of_separable_splitting_field_aux [hFE : FiniteDimensional F E] [sp : p.I
   have h : IsIntegral K x :=
     isIntegral_of_isScalarTower (isIntegral_of_noetherian (IsNoetherian.iff_fg.2 hFE) x)
   have h1 : p ≠ 0 := fun hp => by
-    rw [hp, Polynomial.map_zero, Polynomial.roots_zero] at hx
+    rw [hp, Polynomial.aroots_zero] at hx
     exact Multiset.not_mem_zero x hx
   have h2 : minpoly K x ∣ p.map (algebraMap F K) := by
     apply minpoly.dvd
@@ -396,19 +396,19 @@ theorem of_separable_splitting_field [sp : p.IsSplittingField F E] (hp : p.Separ
     IsGalois F E := by
   haveI hFE : FiniteDimensional F E := Polynomial.IsSplittingField.finiteDimensional E p
   letI := Classical.decEq E
-  let s := (p.map (algebraMap F E)).roots.toFinset
-  have adjoin_root : IntermediateField.adjoin F (s : Set E) = ⊤ := by
+  let s := p.rootSet E
+  have adjoin_root : IntermediateField.adjoin F s = ⊤ := by
     apply IntermediateField.toSubalgebra_injective
     rw [IntermediateField.top_toSubalgebra, ← top_le_iff, ← sp.adjoin_rootSet]
     apply IntermediateField.algebra_adjoin_le_adjoin
   let P : IntermediateField F E → Prop := fun K => Fintype.card (K →ₐ[F] E) = finrank F K
-  suffices P (IntermediateField.adjoin F ↑s) by
+  suffices P (IntermediateField.adjoin F s) by
     rw [adjoin_root] at this
     apply of_card_aut_eq_finrank
     rw [← Eq.trans this (LinearEquiv.finrank_eq IntermediateField.topEquiv.toLinearEquiv)]
     exact Fintype.card_congr ((algEquivEquivAlgHom F E).toEquiv.trans
       (IntermediateField.topEquiv.symm.arrowCongr AlgEquiv.refl))
-  apply IntermediateField.induction_on_adjoin_finset s P
+  apply IntermediateField.induction_on_adjoin_finset _ P
   · have key := IntermediateField.card_algHom_adjoin_integral F (K := E)
       (show IsIntegral F (0 : E) from isIntegral_zero)
     rw [minpoly.zero, Polynomial.natDegree_X] at key

@@ -186,7 +186,6 @@ def FixedPoints.intermediateField (M : Type*) [Monoid M] [MulSemiringAction M E]
     [SMulCommClass M F E] : IntermediateField F E :=
   { FixedPoints.subfield M E with
     carrier := MulAction.fixedPoints M E
-    neg_mem' := fun x => by simp
     algebraMap_mem' := fun a g => by rw [Algebra.algebraMap_eq_smul_one, smul_comm, smul_one] }
 #align fixed_points.intermediate_field FixedPoints.intermediateField
 

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

This has nice performance benefits.

@@ -41,7 +41,7 @@ open FiniteDimensional AlgEquiv
 
 section
 
-variable (F : Type _) [Field F] (E : Type _) [Field E] [Algebra F E]
+variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
 
 /-- A field extension E/F is Galois if it is both separable and normal. Note that in mathlib
 a separable extension of fields is by definition algebraic. -/
@@ -85,7 +85,7 @@ theorem splits [IsGalois F E] (x : E) : (minpoly F x).Splits (algebraMap F E) :=
 
 variable (E)
 
-instance of_fixed_field (G : Type _) [Group G] [Finite G] [MulSemiringAction G E] :
+instance of_fixed_field (G : Type*) [Group G] [Finite G] [MulSemiringAction G E] :
     IsGalois (FixedPoints.subfield G E) E :=
   ⟨⟩
 #align is_galois.of_fixed_field IsGalois.of_fixed_field
@@ -131,7 +131,7 @@ end
 
 section IsGaloisTower
 
-variable (F K E : Type _) [Field F] [Field K] [Field E] {E' : Type _} [Field E'] [Algebra F E']
+variable (F K E : Type*) [Field F] [Field K] [Field E] {E' : Type*} [Field E'] [Algebra F E']
 
 variable [Algebra F K] [Algebra F E] [Algebra K E] [IsScalarTower F K E]
 
@@ -176,13 +176,13 @@ end IsGaloisTower
 
 section GaloisCorrespondence
 
-variable {F : Type _} [Field F] {E : Type _} [Field E] [Algebra F E]
+variable {F : Type*} [Field F] {E : Type*} [Field E] [Algebra F E]
 
 variable (H : Subgroup (E ≃ₐ[F] E)) (K : IntermediateField F E)
 
 /-- The intermediate field of fixed points fixed by a monoid action that commutes with the
 `F`-action on `E`. -/
-def FixedPoints.intermediateField (M : Type _) [Monoid M] [MulSemiringAction M E]
+def FixedPoints.intermediateField (M : Type*) [Monoid M] [MulSemiringAction M E]
     [SMulCommClass M F E] : IntermediateField F E :=
   { FixedPoints.subfield M E with
     carrier := MulAction.fixedPoints M E
@@ -317,7 +317,7 @@ end GaloisCorrespondence
 
 section GaloisEquivalentDefinitions
 
-variable (F : Type _) [Field F] (E : Type _) [Field E] [Algebra F E]
+variable (F : Type*) [Field F] (E : Type*) [Field E] [Algebra F E]
 
 namespace IsGalois
 
@@ -358,7 +358,7 @@ variable {F} {E}
 variable {p : F[X]}
 
 theorem of_separable_splitting_field_aux [hFE : FiniteDimensional F E] [sp : p.IsSplittingField F E]
-    (hp : p.Separable) (K : Type _) [Field K] [Algebra F K] [Algebra K E] [IsScalarTower F K E]
+    (hp : p.Separable) (K : Type*) [Field K] [Algebra F K] [Algebra K E] [IsScalarTower F K E]
     {x : E} (hx : x ∈ (p.map (algebraMap F E)).roots)
     -- these are both implied by `hFE`, but as they carry data this makes the lemma more general
     [Fintype (K →ₐ[F] E)]
@@ -454,7 +454,7 @@ end GaloisEquivalentDefinitions
 
 section normalClosure
 
-variable (k K F : Type _) [Field k] [Field K] [Field F] [Algebra k K] [Algebra k F] [Algebra K F]
+variable (k K F : Type*) [Field k] [Field K] [Field F] [Algebra k K] [Algebra k F] [Algebra K F]
   [IsScalarTower k K F] [IsGalois k F]
 
 instance IsGalois.normalClosure : IsGalois k (normalClosure k K F)
@@ -465,7 +465,7 @@ end normalClosure
 
 section IsAlgClosure
 
-instance (priority := 100) IsAlgClosure.isGalois (k K : Type _) [Field k] [Field K] [Algebra k K]
+instance (priority := 100) IsAlgClosure.isGalois (k K : Type*) [Field k] [Field K] [Algebra k K]
     [IsAlgClosure k K] [CharZero k] : IsGalois k K where
 #align is_alg_closure.is_galois IsAlgClosure.isGalois
 

This PR moves normalClosure to a separate file, in preparation for #6163.

Co-authored-by: Thomas Browning <tb65536@users.noreply.github.com>

@@ -3,8 +3,9 @@ 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.FieldTheory.PrimitiveElement
 import Mathlib.FieldTheory.Fixed
+import Mathlib.FieldTheory.NormalClosure
+import Mathlib.FieldTheory.PrimitiveElement
 import Mathlib.GroupTheory.GroupAction.FixingSubgroup
 
 #align_import field_theory.galois from "leanprover-community/mathlib"@"9fb8964792b4237dac6200193a0d533f1b3f7423"

• depends on: #5966

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

@@ -2,16 +2,13 @@
 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.galois
-! leanprover-community/mathlib commit 9fb8964792b4237dac6200193a0d533f1b3f7423
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.FieldTheory.PrimitiveElement
 import Mathlib.FieldTheory.Fixed
 import Mathlib.GroupTheory.GroupAction.FixingSubgroup
 
+#align_import field_theory.galois from "leanprover-community/mathlib"@"9fb8964792b4237dac6200193a0d533f1b3f7423"
+
 /-!
 # Galois Extensions
 

Co-authored-by: Jeremy Tan Jie Rui <reddeloostw@gmail.com>