field_theory.finite.galois_field ⟷ Mathlib.FieldTheory.Finite.GaloisField

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Aaron Anderson, Alex J. Best, Johan Commelin, Eric Rodriguez, Ruben Van de Velde
 -/
 import Algebra.CharP.Algebra
-import Data.Zmod.Algebra
+import Data.ZMod.Algebra
 import FieldTheory.Finite.Basic
 import FieldTheory.Galois
 import FieldTheory.SplittingField.IsSplittingField

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

@@ -62,7 +62,7 @@ instance FiniteField.isSplittingField_sub (K F : Type _) [Field K] [Fintype K] [
 theorem galois_poly_separable {K : Type _} [Field K] (p q : ℕ) [CharP K p] (h : p ∣ q) :
     Separable (X ^ q - X : K[X]) := by
   use 1, X ^ q - X - 1
-  rw [← CharP.cast_eq_zero_iff K[X] p] at h 
+  rw [← CharP.cast_eq_zero_iff K[X] p] at h
   rw [derivative_sub, derivative_X_pow, derivative_X, C_eq_nat_cast, h]
   ring
 #align galois_poly_separable galois_poly_separable
@@ -110,11 +110,11 @@ theorem finrank {n} (h : n ≠ 0) : FiniteDimensional.finrank (ZMod p) (GaloisFi
       (splitting_field.splits g_poly)
   have nat_degree_eq : g_poly.natDegree = p ^ n :=
     FiniteField.X_pow_card_pow_sub_X_natDegree_eq _ h hp
-  rw [nat_degree_eq] at key 
+  rw [nat_degree_eq] at key
   suffices g_poly.rootSet (GaloisField p n) = Set.univ
     by
-    simp_rw [this, ← Fintype.ofEquiv_card (Equiv.Set.univ _)] at key 
-    rw [@card_eq_pow_finrank (ZMod p), ZMod.card] at key 
+    simp_rw [this, ← Fintype.ofEquiv_card (Equiv.Set.univ _)] at key
+    rw [@card_eq_pow_finrank (ZMod p), ZMod.card] at key
     exact Nat.pow_right_injective (Nat.Prime.one_lt' p).out key
   rw [Set.eq_univ_iff_forall]
   suffices
@@ -131,8 +131,8 @@ theorem finrank {n} (h : n ≠ 0) : FiniteDimensional.finrank (ZMod p) (GaloisFi
   · rintro x (⟨r, rfl⟩ | hx)
     · simp only [aeval_X_pow, aeval_X, AlgHom.map_sub]
       rw [← map_pow, ZMod.pow_card_pow, sub_self]
-    · dsimp only [GaloisField] at hx 
-      rwa [mem_root_set_of_ne aux] at hx ; infer_instance
+    · dsimp only [GaloisField] at hx
+      rwa [mem_root_set_of_ne aux] at hx; infer_instance
   · dsimp only [g_poly]
     rw [← coeff_zero_eq_aeval_zero']
     simp only [coeff_X_pow, coeff_X_zero, sub_zero, _root_.map_eq_zero, ite_eq_right_iff,
@@ -237,7 +237,7 @@ def algEquivOfCardEq (p : ℕ) [Fact p.Prime] [Algebra (ZMod p) K] [Algebra (ZMo
   have : CharP K' p := by rw [← Algebra.charP_iff (ZMod p) K' p]; exact ZMod.charP p
   choose n a hK using FiniteField.card K p
   choose n' a' hK' using FiniteField.card K' p
-  rw [hK, hK'] at hKK' 
+  rw [hK, hK'] at hKK'
   have hGalK := GaloisField.algEquivGaloisField p n hK
   have hK'Gal := (GaloisField.algEquivGaloisField p n' hK').symm
   rw [Nat.pow_right_injective (Fact.out (Nat.Prime p)).one_lt hKK'] at *
@@ -260,7 +260,7 @@ def ringEquivOfCardEq (hKK' : Fintype.card K = Fintype.card K') : K ≃+* K' :=
     -- := eq_prime_of_eq_prime_pow
     by_contra hne
     have h2 := Nat.coprime_pow_primes n n' hp hp' hne
-    rw [(Eq.congr hK hK').mp hKK', Nat.coprime_self, pow_eq_one_iff (PNat.ne_zero n')] at h2 
+    rw [(Eq.congr hK hK').mp hKK', Nat.coprime_self, pow_eq_one_iff (PNat.ne_zero n')] at h2
     exact Nat.Prime.ne_one hp' h2
     all_goals infer_instance
   rw [← hpp'] at *

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

@@ -50,7 +50,11 @@ instance FiniteField.isSplittingField_sub (K F : Type _) [Field K] [Fintype K] [
       FiniteField.X_pow_card_sub_X_natDegree_eq K Fintype.one_lt_card
     rw [← splits_id_iff_splits, splits_iff_card_roots, Polynomial.map_sub, Polynomial.map_pow,
       map_X, h, FiniteField.roots_X_pow_card_sub_X K, ← Finset.card_def, Finset.card_univ]
-  adjoin_rootSet := by classical
+  adjoin_rootSet := by
+    classical
+    trans Algebra.adjoin F ((roots (X ^ Fintype.card K - X : K[X])).toFinset : Set K)
+    · simp only [root_set, Polynomial.map_pow, map_X, Polynomial.map_sub]
+    · rw [FiniteField.roots_X_pow_card_sub_X, val_to_finset, coe_univ, Algebra.adjoin_univ]
 #align finite_field.has_sub.sub.polynomial.is_splitting_field FiniteField.isSplittingField_sub
 -/
 

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

@@ -50,11 +50,7 @@ instance FiniteField.isSplittingField_sub (K F : Type _) [Field K] [Fintype K] [
       FiniteField.X_pow_card_sub_X_natDegree_eq K Fintype.one_lt_card
     rw [← splits_id_iff_splits, splits_iff_card_roots, Polynomial.map_sub, Polynomial.map_pow,
       map_X, h, FiniteField.roots_X_pow_card_sub_X K, ← Finset.card_def, Finset.card_univ]
-  adjoin_rootSet := by
-    classical
-    trans Algebra.adjoin F ((roots (X ^ Fintype.card K - X : K[X])).toFinset : Set K)
-    · simp only [root_set, Polynomial.map_pow, map_X, Polynomial.map_sub]
-    · rw [FiniteField.roots_X_pow_card_sub_X, val_to_finset, coe_univ, Algebra.adjoin_univ]
+  adjoin_rootSet := by classical
 #align finite_field.has_sub.sub.polynomial.is_splitting_field FiniteField.isSplittingField_sub
 -/
 

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

@@ -3,11 +3,11 @@ Copyright (c) 2021 Johan Commelin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Aaron Anderson, Alex J. Best, Johan Commelin, Eric Rodriguez, Ruben Van de Velde
 -/
-import Mathbin.Algebra.CharP.Algebra
-import Mathbin.Data.Zmod.Algebra
-import Mathbin.FieldTheory.Finite.Basic
-import Mathbin.FieldTheory.Galois
-import Mathbin.FieldTheory.SplittingField.IsSplittingField
+import Algebra.CharP.Algebra
+import Data.Zmod.Algebra
+import FieldTheory.Finite.Basic
+import FieldTheory.Galois
+import FieldTheory.SplittingField.IsSplittingField
 
 #align_import field_theory.finite.galois_field from "leanprover-community/mathlib"@"1b089e3bdc3ce6b39cd472543474a0a137128c6c"
 

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

@@ -2,11 +2,6 @@
 Copyright (c) 2021 Johan Commelin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Aaron Anderson, Alex J. Best, Johan Commelin, Eric Rodriguez, Ruben Van de Velde
-
-! This file was ported from Lean 3 source module field_theory.finite.galois_field
-! 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.Algebra.CharP.Algebra
 import Mathbin.Data.Zmod.Algebra
@@ -14,6 +9,8 @@ import Mathbin.FieldTheory.Finite.Basic
 import Mathbin.FieldTheory.Galois
 import Mathbin.FieldTheory.SplittingField.IsSplittingField
 
+#align_import field_theory.finite.galois_field from "leanprover-community/mathlib"@"1b089e3bdc3ce6b39cd472543474a0a137128c6c"
+
 /-!
 # Galois fields
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/93f880918cb51905fd51b76add8273cbc27718ab

@@ -43,9 +43,9 @@ open Polynomial Finset
 
 open scoped Polynomial
 
-#print FiniteField.HasSub.Sub.Polynomial.isSplittingField /-
-instance FiniteField.HasSub.Sub.Polynomial.isSplittingField (K F : Type _) [Field K] [Fintype K]
-    [Field F] [Algebra F K] : IsSplittingField F K (X ^ Fintype.card K - X)
+#print FiniteField.isSplittingField_sub /-
+instance FiniteField.isSplittingField_sub (K F : Type _) [Field K] [Fintype K] [Field F]
+    [Algebra F K] : IsSplittingField F K (X ^ Fintype.card K - X)
     where
   Splits :=
     by
@@ -58,7 +58,7 @@ instance FiniteField.HasSub.Sub.Polynomial.isSplittingField (K F : Type _) [Fiel
     trans Algebra.adjoin F ((roots (X ^ Fintype.card K - X : K[X])).toFinset : Set K)
     · simp only [root_set, Polynomial.map_pow, map_X, Polynomial.map_sub]
     · rw [FiniteField.roots_X_pow_card_sub_X, val_to_finset, coe_univ, Algebra.adjoin_univ]
-#align finite_field.has_sub.sub.polynomial.is_splitting_field FiniteField.HasSub.Sub.Polynomial.isSplittingField
+#align finite_field.has_sub.sub.polynomial.is_splitting_field FiniteField.isSplittingField_sub
 -/
 
 #print galois_poly_separable /-
@@ -164,8 +164,8 @@ theorem card (h : n ≠ 0) : Fintype.card (GaloisField p n) = p ^ n :=
 #align galois_field.card GaloisField.card
 -/
 
-#print GaloisField.splits_zMod_x_pow_sub_x /-
-theorem splits_zMod_x_pow_sub_x : Splits (RingHom.id (ZMod p)) (X ^ p - X) :=
+#print GaloisField.splits_zmod_X_pow_sub_X /-
+theorem splits_zmod_X_pow_sub_X : Splits (RingHom.id (ZMod p)) (X ^ p - X) :=
   by
   have hp : 1 < p := (Fact.out (Nat.Prime p)).one_lt
   have h1 : roots (X ^ p - X : (ZMod p)[X]) = finset.univ.val :=
@@ -177,7 +177,7 @@ theorem splits_zMod_x_pow_sub_x : Splits (RingHom.id (ZMod p)) (X ^ p - X) :=
   cases p;
   cases hp
   rw [splits_iff_card_roots, h1, ← Finset.card_def, Finset.card_univ, h2, ZMod.card]
-#align galois_field.splits_zmod_X_pow_sub_X GaloisField.splits_zMod_x_pow_sub_x
+#align galois_field.splits_zmod_X_pow_sub_X GaloisField.splits_zmod_X_pow_sub_X
 -/
 
 #print GaloisField.equivZmodP /-
@@ -191,16 +191,16 @@ def equivZmodP : GaloisField p 1 ≃ₐ[ZMod p] ZMod p :=
 
 variable {K : Type _} [Field K] [Fintype K] [Algebra (ZMod p) K]
 
-#print GaloisField.splits_x_pow_card_sub_x /-
-theorem splits_x_pow_card_sub_x : Splits (algebraMap (ZMod p) K) (X ^ Fintype.card K - X) :=
-  (FiniteField.HasSub.Sub.Polynomial.isSplittingField K (ZMod p)).Splits
-#align galois_field.splits_X_pow_card_sub_X GaloisField.splits_x_pow_card_sub_x
+#print GaloisField.splits_X_pow_card_sub_X /-
+theorem splits_X_pow_card_sub_X : Splits (algebraMap (ZMod p) K) (X ^ Fintype.card K - X) :=
+  (FiniteField.isSplittingField_sub K (ZMod p)).Splits
+#align galois_field.splits_X_pow_card_sub_X GaloisField.splits_X_pow_card_sub_X
 -/
 
 #print GaloisField.isSplittingField_of_card_eq /-
 theorem isSplittingField_of_card_eq (h : Fintype.card K = p ^ n) :
     IsSplittingField (ZMod p) K (X ^ p ^ n - X) :=
-  h ▸ FiniteField.HasSub.Sub.Polynomial.isSplittingField K (ZMod p)
+  h ▸ FiniteField.isSplittingField_sub K (ZMod p)
 #align galois_field.is_splitting_field_of_card_eq GaloisField.isSplittingField_of_card_eq
 -/
 

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: Aaron Anderson, Alex J. Best, Johan Commelin, Eric Rodriguez, Ruben Van de Velde
 
 ! This file was ported from Lean 3 source module field_theory.finite.galois_field
-! leanprover-community/mathlib commit 0723536a0522d24fc2f159a096fb3304bef77472
+! leanprover-community/mathlib commit 1b089e3bdc3ce6b39cd472543474a0a137128c6c
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -17,6 +17,9 @@ import Mathbin.FieldTheory.SplittingField.IsSplittingField
 /-!
 # Galois fields
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 If `p` is a prime number, and `n` a natural number,
 then `galois_field p n` is defined as the splitting field of `X^(p^n) - X` over `zmod p`.
 It is a finite field with `p ^ n` elements.

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

@@ -40,6 +40,7 @@ open Polynomial Finset
 
 open scoped Polynomial
 
+#print FiniteField.HasSub.Sub.Polynomial.isSplittingField /-
 instance FiniteField.HasSub.Sub.Polynomial.isSplittingField (K F : Type _) [Field K] [Fintype K]
     [Field F] [Algebra F K] : IsSplittingField F K (X ^ Fintype.card K - X)
     where
@@ -55,7 +56,9 @@ instance FiniteField.HasSub.Sub.Polynomial.isSplittingField (K F : Type _) [Fiel
     · simp only [root_set, Polynomial.map_pow, map_X, Polynomial.map_sub]
     · rw [FiniteField.roots_X_pow_card_sub_X, val_to_finset, coe_univ, Algebra.adjoin_univ]
 #align finite_field.has_sub.sub.polynomial.is_splitting_field FiniteField.HasSub.Sub.Polynomial.isSplittingField
+-/
 
+#print galois_poly_separable /-
 theorem galois_poly_separable {K : Type _} [Field K] (p q : ℕ) [CharP K p] (h : p ∣ q) :
     Separable (X ^ q - X : K[X]) := by
   use 1, X ^ q - X - 1
@@ -63,7 +66,9 @@ theorem galois_poly_separable {K : Type _} [Field K] (p q : ℕ) [CharP K p] (h
   rw [derivative_sub, derivative_X_pow, derivative_X, C_eq_nat_cast, h]
   ring
 #align galois_poly_separable galois_poly_separable
+-/
 
+#print GaloisField /-
 /-- A finite field with `p ^ n` elements.
 Every field with the same cardinality is (non-canonically)
 isomorphic to this field. -/
@@ -71,6 +76,7 @@ def GaloisField (p : ℕ) [Fact p.Prime] (n : ℕ) :=
   SplittingField (X ^ p ^ n - X : (ZMod p)[X])
 deriving Field
 #align galois_field GaloisField
+-/
 
 instance : Inhabited (@GaloisField 2 (Fact.mk Nat.prime_two) 1) :=
   ⟨37⟩
@@ -93,6 +99,7 @@ instance : Fintype (GaloisField p n) :=
   dsimp only [GaloisField]
   exact FiniteDimensional.fintypeOfFintype (ZMod p) (GaloisField p n)
 
+#print GaloisField.finrank /-
 theorem finrank {n} (h : n ≠ 0) : FiniteDimensional.finrank (ZMod p) (GaloisField p n) = n :=
   by
   set g_poly := (X ^ p ^ n - X : (ZMod p)[X])
@@ -144,13 +151,17 @@ theorem finrank {n} (h : n ≠ 0) : FiniteDimensional.finrank (ZMod p) (GaloisFi
     intro x y hx hy
     rw [hx, hy]
 #align galois_field.finrank GaloisField.finrank
+-/
 
+#print GaloisField.card /-
 theorem card (h : n ≠ 0) : Fintype.card (GaloisField p n) = p ^ n :=
   by
   let b := IsNoetherian.finsetBasis (ZMod p) (GaloisField p n)
   rw [Module.card_fintype b, ← FiniteDimensional.finrank_eq_card_basis b, ZMod.card, finrank p h]
 #align galois_field.card GaloisField.card
+-/
 
+#print GaloisField.splits_zMod_x_pow_sub_x /-
 theorem splits_zMod_x_pow_sub_x : Splits (RingHom.id (ZMod p)) (X ^ p - X) :=
   by
   have hp : 1 < p := (Fact.out (Nat.Prime p)).one_lt
@@ -164,24 +175,31 @@ theorem splits_zMod_x_pow_sub_x : Splits (RingHom.id (ZMod p)) (X ^ p - X) :=
   cases hp
   rw [splits_iff_card_roots, h1, ← Finset.card_def, Finset.card_univ, h2, ZMod.card]
 #align galois_field.splits_zmod_X_pow_sub_X GaloisField.splits_zMod_x_pow_sub_x
+-/
 
+#print GaloisField.equivZmodP /-
 /-- A Galois field with exponent 1 is equivalent to `zmod` -/
 def equivZmodP : GaloisField p 1 ≃ₐ[ZMod p] ZMod p :=
   let h : (X ^ p ^ 1 : (ZMod p)[X]) = X ^ Fintype.card (ZMod p) := by rw [pow_one, ZMod.card p]
   let inst : IsSplittingField (ZMod p) (ZMod p) (X ^ p ^ 1 - X) := by rw [h]; infer_instance
   (@IsSplittingField.algEquiv _ (ZMod p) _ _ _ (X ^ p ^ 1 - X : (ZMod p)[X]) inst).symm
 #align galois_field.equiv_zmod_p GaloisField.equivZmodP
+-/
 
 variable {K : Type _} [Field K] [Fintype K] [Algebra (ZMod p) K]
 
+#print GaloisField.splits_x_pow_card_sub_x /-
 theorem splits_x_pow_card_sub_x : Splits (algebraMap (ZMod p) K) (X ^ Fintype.card K - X) :=
   (FiniteField.HasSub.Sub.Polynomial.isSplittingField K (ZMod p)).Splits
 #align galois_field.splits_X_pow_card_sub_X GaloisField.splits_x_pow_card_sub_x
+-/
 
+#print GaloisField.isSplittingField_of_card_eq /-
 theorem isSplittingField_of_card_eq (h : Fintype.card K = p ^ n) :
     IsSplittingField (ZMod p) K (X ^ p ^ n - X) :=
   h ▸ FiniteField.HasSub.Sub.Polynomial.isSplittingField K (ZMod p)
 #align galois_field.is_splitting_field_of_card_eq GaloisField.isSplittingField_of_card_eq
+-/
 
 instance (priority := 100) {K K' : Type _} [Field K] [Field K'] [Finite K'] [Algebra K K'] :
     IsGalois K K' := by
@@ -195,11 +213,13 @@ instance (priority := 100) {K K' : Type _} [Field K] [Field K'] [Finite K'] [Alg
         (let ⟨n, hp, hn⟩ := FiniteField.card K' p
         hn.symm ▸ dvd_pow_self p n.NeZero))
 
+#print GaloisField.algEquivGaloisField /-
 /-- Any finite field is (possibly non canonically) isomorphic to some Galois field. -/
 def algEquivGaloisField (h : Fintype.card K = p ^ n) : K ≃ₐ[ZMod p] GaloisField p n :=
   haveI := is_splitting_field_of_card_eq _ _ h
   is_splitting_field.alg_equiv _ _
 #align galois_field.alg_equiv_galois_field GaloisField.algEquivGaloisField
+-/
 
 end GaloisField
 
@@ -207,6 +227,7 @@ namespace FiniteField
 
 variable {K : Type _} [Field K] [Fintype K] {K' : Type _} [Field K'] [Fintype K']
 
+#print FiniteField.algEquivOfCardEq /-
 /-- Uniqueness of finite fields:
   Any two finite fields of the same cardinality are (possibly non canonically) isomorphic-/
 def algEquivOfCardEq (p : ℕ) [Fact p.Prime] [Algebra (ZMod p) K] [Algebra (ZMod p) K']
@@ -222,7 +243,9 @@ def algEquivOfCardEq (p : ℕ) [Fact p.Prime] [Algebra (ZMod p) K] [Algebra (ZMo
   rw [Nat.pow_right_injective (Fact.out (Nat.Prime p)).one_lt hKK'] at *
   use AlgEquiv.trans hGalK hK'Gal
 #align finite_field.alg_equiv_of_card_eq FiniteField.algEquivOfCardEq
+-/
 
+#print FiniteField.ringEquivOfCardEq /-
 /-- Uniqueness of finite fields:
   Any two finite fields of the same cardinality are (possibly non canonically) isomorphic-/
 def ringEquivOfCardEq (hKK' : Fintype.card K = Fintype.card K') : K ≃+* K' :=
@@ -246,6 +269,7 @@ def ringEquivOfCardEq (hKK' : Fintype.card K = Fintype.card K') : K ≃+* K' :=
   letI : Algebra (ZMod p) K' := ZMod.algebra _ _
   exact alg_equiv_of_card_eq p hKK'
 #align finite_field.ring_equiv_of_card_eq FiniteField.ringEquivOfCardEq
+-/
 
 end FiniteField
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/0723536a0522d24fc2f159a096fb3304bef77472

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Aaron Anderson, Alex J. Best, Johan Commelin, Eric Rodriguez, Ruben Van de Velde
 
 ! This file was ported from Lean 3 source module field_theory.finite.galois_field
-! leanprover-community/mathlib commit 4b05d3f4f0601dca8abf99c4ec99187682ed0bba
+! leanprover-community/mathlib commit 0723536a0522d24fc2f159a096fb3304bef77472
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -165,8 +165,6 @@ theorem splits_zMod_x_pow_sub_x : Splits (RingHom.id (ZMod p)) (X ^ p - X) :=
   rw [splits_iff_card_roots, h1, ← Finset.card_def, Finset.card_univ, h2, ZMod.card]
 #align galois_field.splits_zmod_X_pow_sub_X GaloisField.splits_zMod_x_pow_sub_x
 
-attribute [-instance] ZMod.algebra
-
 /-- A Galois field with exponent 1 is equivalent to `zmod` -/
 def equivZmodP : GaloisField p 1 ≃ₐ[ZMod p] ZMod p :=
   let h : (X ^ p ^ 1 : (ZMod p)[X]) = X ^ Fintype.card (ZMod p) := by rw [pow_one, ZMod.card p]
@@ -244,6 +242,8 @@ def ringEquivOfCardEq (hKK' : Fintype.card K = Fintype.card K') : K ≃+* K' :=
     all_goals infer_instance
   rw [← hpp'] at *
   haveI := fact_iff.2 hp
+  letI : Algebra (ZMod p) K := ZMod.algebra _ _
+  letI : Algebra (ZMod p) K' := ZMod.algebra _ _
   exact alg_equiv_of_card_eq p hKK'
 #align finite_field.ring_equiv_of_card_eq FiniteField.ringEquivOfCardEq
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Aaron Anderson, Alex J. Best, Johan Commelin, Eric Rodriguez, Ruben Van de Velde
 
 ! This file was ported from Lean 3 source module field_theory.finite.galois_field
-! leanprover-community/mathlib commit 9fb8964792b4237dac6200193a0d533f1b3f7423
+! leanprover-community/mathlib commit 4b05d3f4f0601dca8abf99c4ec99187682ed0bba
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -165,11 +165,13 @@ theorem splits_zMod_x_pow_sub_x : Splits (RingHom.id (ZMod p)) (X ^ p - X) :=
   rw [splits_iff_card_roots, h1, ← Finset.card_def, Finset.card_univ, h2, ZMod.card]
 #align galois_field.splits_zmod_X_pow_sub_X GaloisField.splits_zMod_x_pow_sub_x
 
+attribute [-instance] ZMod.algebra
+
 /-- A Galois field with exponent 1 is equivalent to `zmod` -/
 def equivZmodP : GaloisField p 1 ≃ₐ[ZMod p] ZMod p :=
-  have h : (X ^ p ^ 1 : (ZMod p)[X]) = X ^ Fintype.card (ZMod p) := by rw [pow_one, ZMod.card p]
-  have inst : IsSplittingField (ZMod p) (ZMod p) (X ^ p ^ 1 - X) := by rw [h]; infer_instance
-  (is_splitting_field.alg_equiv (ZMod p) (X ^ p ^ 1 - X : (ZMod p)[X])).symm
+  let h : (X ^ p ^ 1 : (ZMod p)[X]) = X ^ Fintype.card (ZMod p) := by rw [pow_one, ZMod.card p]
+  let inst : IsSplittingField (ZMod p) (ZMod p) (X ^ p ^ 1 - X) := by rw [h]; infer_instance
+  (@IsSplittingField.algEquiv _ (ZMod p) _ _ _ (X ^ p ^ 1 - X : (ZMod p)[X]) inst).symm
 #align galois_field.equiv_zmod_p GaloisField.equivZmodP
 
 variable {K : Type _} [Field K] [Fintype K] [Algebra (ZMod p) K]

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: Aaron Anderson, Alex J. Best, Johan Commelin, Eric Rodriguez, Ruben Van de Velde
 
 ! This file was ported from Lean 3 source module field_theory.finite.galois_field
-! leanprover-community/mathlib commit df76f43357840485b9d04ed5dee5ab115d420e87
+! leanprover-community/mathlib commit 9fb8964792b4237dac6200193a0d533f1b3f7423
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -49,10 +49,10 @@ instance FiniteField.HasSub.Sub.Polynomial.isSplittingField (K F : Type _) [Fiel
       FiniteField.X_pow_card_sub_X_natDegree_eq K Fintype.one_lt_card
     rw [← splits_id_iff_splits, splits_iff_card_roots, Polynomial.map_sub, Polynomial.map_pow,
       map_X, h, FiniteField.roots_X_pow_card_sub_X K, ← Finset.card_def, Finset.card_univ]
-  adjoin_roots := by
+  adjoin_rootSet := by
     classical
     trans Algebra.adjoin F ((roots (X ^ Fintype.card K - X : K[X])).toFinset : Set K)
-    · simp only [Polynomial.map_pow, map_X, Polynomial.map_sub]
+    · simp only [root_set, Polynomial.map_pow, map_X, Polynomial.map_sub]
     · rw [FiniteField.roots_X_pow_card_sub_X, val_to_finset, coe_univ, Algebra.adjoin_univ]
 #align finite_field.has_sub.sub.polynomial.is_splitting_field FiniteField.HasSub.Sub.Polynomial.isSplittingField
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/a3209ddf94136d36e5e5c624b10b2a347cc9d090

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Aaron Anderson, Alex J. Best, Johan Commelin, Eric Rodriguez, Ruben Van de Velde
 
 ! This file was ported from Lean 3 source module field_theory.finite.galois_field
-! leanprover-community/mathlib commit 12a85fac627bea918960da036049d611b1a3ee43
+! leanprover-community/mathlib commit df76f43357840485b9d04ed5dee5ab115d420e87
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -12,7 +12,7 @@ import Mathbin.Algebra.CharP.Algebra
 import Mathbin.Data.Zmod.Algebra
 import Mathbin.FieldTheory.Finite.Basic
 import Mathbin.FieldTheory.Galois
-import Mathbin.FieldTheory.SplittingField
+import Mathbin.FieldTheory.SplittingField.IsSplittingField
 
 /-!
 # Galois fields

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

@@ -46,14 +46,14 @@ instance FiniteField.HasSub.Sub.Polynomial.isSplittingField (K F : Type _) [Fiel
   Splits :=
     by
     have h : (X ^ Fintype.card K - X : K[X]).natDegree = Fintype.card K :=
-      FiniteField.x_pow_card_sub_x_natDegree_eq K Fintype.one_lt_card
+      FiniteField.X_pow_card_sub_X_natDegree_eq K Fintype.one_lt_card
     rw [← splits_id_iff_splits, splits_iff_card_roots, Polynomial.map_sub, Polynomial.map_pow,
-      map_X, h, FiniteField.roots_x_pow_card_sub_x K, ← Finset.card_def, Finset.card_univ]
+      map_X, h, FiniteField.roots_X_pow_card_sub_X K, ← Finset.card_def, Finset.card_univ]
   adjoin_roots := by
     classical
-      trans Algebra.adjoin F ((roots (X ^ Fintype.card K - X : K[X])).toFinset : Set K)
-      · simp only [Polynomial.map_pow, map_X, Polynomial.map_sub]
-      · rw [FiniteField.roots_x_pow_card_sub_x, val_to_finset, coe_univ, Algebra.adjoin_univ]
+    trans Algebra.adjoin F ((roots (X ^ Fintype.card K - X : K[X])).toFinset : Set K)
+    · simp only [Polynomial.map_pow, map_X, Polynomial.map_sub]
+    · rw [FiniteField.roots_X_pow_card_sub_X, val_to_finset, coe_univ, Algebra.adjoin_univ]
 #align finite_field.has_sub.sub.polynomial.is_splitting_field FiniteField.HasSub.Sub.Polynomial.isSplittingField
 
 theorem galois_poly_separable {K : Type _} [Field K] (p q : ℕ) [CharP K p] (h : p ∣ q) :
@@ -97,12 +97,12 @@ theorem finrank {n} (h : n ≠ 0) : FiniteDimensional.finrank (ZMod p) (GaloisFi
   by
   set g_poly := (X ^ p ^ n - X : (ZMod p)[X])
   have hp : 1 < p := (Fact.out (Nat.Prime p)).one_lt
-  have aux : g_poly ≠ 0 := FiniteField.x_pow_card_pow_sub_x_ne_zero _ h hp
+  have aux : g_poly ≠ 0 := FiniteField.X_pow_card_pow_sub_X_ne_zero _ h hp
   have key : Fintype.card (g_poly.rootSet (GaloisField p n)) = g_poly.natDegree :=
     card_root_set_eq_nat_degree (galois_poly_separable p _ (dvd_pow (dvd_refl p) h))
       (splitting_field.splits g_poly)
   have nat_degree_eq : g_poly.natDegree = p ^ n :=
-    FiniteField.x_pow_card_pow_sub_x_natDegree_eq _ h hp
+    FiniteField.X_pow_card_pow_sub_X_natDegree_eq _ h hp
   rw [nat_degree_eq] at key 
   suffices g_poly.rootSet (GaloisField p n) = Set.univ
     by
@@ -156,9 +156,9 @@ theorem splits_zMod_x_pow_sub_x : Splits (RingHom.id (ZMod p)) (X ^ p - X) :=
   have hp : 1 < p := (Fact.out (Nat.Prime p)).one_lt
   have h1 : roots (X ^ p - X : (ZMod p)[X]) = finset.univ.val :=
     by
-    convert FiniteField.roots_x_pow_card_sub_x _
+    convert FiniteField.roots_X_pow_card_sub_X _
     exact (ZMod.card p).symm
-  have h2 := FiniteField.x_pow_card_sub_x_natDegree_eq (ZMod p) hp
+  have h2 := FiniteField.X_pow_card_sub_X_natDegree_eq (ZMod p) hp
   -- We discharge the `p = 0` separately, to avoid typeclass issues on `zmod p`.
   cases p;
   cases hp

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

@@ -59,7 +59,7 @@ instance FiniteField.HasSub.Sub.Polynomial.isSplittingField (K F : Type _) [Fiel
 theorem galois_poly_separable {K : Type _} [Field K] (p q : ℕ) [CharP K p] (h : p ∣ q) :
     Separable (X ^ q - X : K[X]) := by
   use 1, X ^ q - X - 1
-  rw [← CharP.cast_eq_zero_iff K[X] p] at h
+  rw [← CharP.cast_eq_zero_iff K[X] p] at h 
   rw [derivative_sub, derivative_X_pow, derivative_X, C_eq_nat_cast, h]
   ring
 #align galois_poly_separable galois_poly_separable
@@ -68,7 +68,8 @@ theorem galois_poly_separable {K : Type _} [Field K] (p q : ℕ) [CharP K p] (h
 Every field with the same cardinality is (non-canonically)
 isomorphic to this field. -/
 def GaloisField (p : ℕ) [Fact p.Prime] (n : ℕ) :=
-  SplittingField (X ^ p ^ n - X : (ZMod p)[X])deriving Field
+  SplittingField (X ^ p ^ n - X : (ZMod p)[X])
+deriving Field
 #align galois_field GaloisField
 
 instance : Inhabited (@GaloisField 2 (Fact.mk Nat.prime_two) 1) :=
@@ -102,11 +103,11 @@ theorem finrank {n} (h : n ≠ 0) : FiniteDimensional.finrank (ZMod p) (GaloisFi
       (splitting_field.splits g_poly)
   have nat_degree_eq : g_poly.natDegree = p ^ n :=
     FiniteField.x_pow_card_pow_sub_x_natDegree_eq _ h hp
-  rw [nat_degree_eq] at key
+  rw [nat_degree_eq] at key 
   suffices g_poly.rootSet (GaloisField p n) = Set.univ
     by
-    simp_rw [this, ← Fintype.ofEquiv_card (Equiv.Set.univ _)] at key
-    rw [@card_eq_pow_finrank (ZMod p), ZMod.card] at key
+    simp_rw [this, ← Fintype.ofEquiv_card (Equiv.Set.univ _)] at key 
+    rw [@card_eq_pow_finrank (ZMod p), ZMod.card] at key 
     exact Nat.pow_right_injective (Nat.Prime.one_lt' p).out key
   rw [Set.eq_univ_iff_forall]
   suffices
@@ -123,8 +124,8 @@ theorem finrank {n} (h : n ≠ 0) : FiniteDimensional.finrank (ZMod p) (GaloisFi
   · rintro x (⟨r, rfl⟩ | hx)
     · simp only [aeval_X_pow, aeval_X, AlgHom.map_sub]
       rw [← map_pow, ZMod.pow_card_pow, sub_self]
-    · dsimp only [GaloisField] at hx
-      rwa [mem_root_set_of_ne aux] at hx; infer_instance
+    · dsimp only [GaloisField] at hx 
+      rwa [mem_root_set_of_ne aux] at hx ; infer_instance
   · dsimp only [g_poly]
     rw [← coeff_zero_eq_aeval_zero']
     simp only [coeff_X_pow, coeff_X_zero, sub_zero, _root_.map_eq_zero, ite_eq_right_iff,
@@ -215,7 +216,7 @@ def algEquivOfCardEq (p : ℕ) [Fact p.Prime] [Algebra (ZMod p) K] [Algebra (ZMo
   have : CharP K' p := by rw [← Algebra.charP_iff (ZMod p) K' p]; exact ZMod.charP p
   choose n a hK using FiniteField.card K p
   choose n' a' hK' using FiniteField.card K' p
-  rw [hK, hK'] at hKK'
+  rw [hK, hK'] at hKK' 
   have hGalK := GaloisField.algEquivGaloisField p n hK
   have hK'Gal := (GaloisField.algEquivGaloisField p n' hK').symm
   rw [Nat.pow_right_injective (Fact.out (Nat.Prime p)).one_lt hKK'] at *
@@ -236,7 +237,7 @@ def ringEquivOfCardEq (hKK' : Fintype.card K = Fintype.card K') : K ≃+* K' :=
     -- := eq_prime_of_eq_prime_pow
     by_contra hne
     have h2 := Nat.coprime_pow_primes n n' hp hp' hne
-    rw [(Eq.congr hK hK').mp hKK', Nat.coprime_self, pow_eq_one_iff (PNat.ne_zero n')] at h2
+    rw [(Eq.congr hK hK').mp hKK', Nat.coprime_self, pow_eq_one_iff (PNat.ne_zero n')] at h2 
     exact Nat.Prime.ne_one hp' h2
     all_goals infer_instance
   rw [← hpp'] at *

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Aaron Anderson, Alex J. Best, Johan Commelin, Eric Rodriguez, Ruben Van de Velde
 
 ! This file was ported from Lean 3 source module field_theory.finite.galois_field
-! leanprover-community/mathlib commit bbeb185db4ccee8ed07dc48449414ebfa39cb821
+! leanprover-community/mathlib commit 12a85fac627bea918960da036049d611b1a3ee43
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -12,6 +12,7 @@ import Mathbin.Algebra.CharP.Algebra
 import Mathbin.Data.Zmod.Algebra
 import Mathbin.FieldTheory.Finite.Basic
 import Mathbin.FieldTheory.Galois
+import Mathbin.FieldTheory.SplittingField
 
 /-!
 # Galois fields
@@ -35,10 +36,26 @@ It is a finite field with `p ^ n` elements.
 
 noncomputable section
 
-open Polynomial
+open Polynomial Finset
 
 open scoped Polynomial
 
+instance FiniteField.HasSub.Sub.Polynomial.isSplittingField (K F : Type _) [Field K] [Fintype K]
+    [Field F] [Algebra F K] : IsSplittingField F K (X ^ Fintype.card K - X)
+    where
+  Splits :=
+    by
+    have h : (X ^ Fintype.card K - X : K[X]).natDegree = Fintype.card K :=
+      FiniteField.x_pow_card_sub_x_natDegree_eq K Fintype.one_lt_card
+    rw [← splits_id_iff_splits, splits_iff_card_roots, Polynomial.map_sub, Polynomial.map_pow,
+      map_X, h, FiniteField.roots_x_pow_card_sub_x K, ← Finset.card_def, Finset.card_univ]
+  adjoin_roots := by
+    classical
+      trans Algebra.adjoin F ((roots (X ^ Fintype.card K - X : K[X])).toFinset : Set K)
+      · simp only [Polynomial.map_pow, map_X, Polynomial.map_sub]
+      · rw [FiniteField.roots_x_pow_card_sub_x, val_to_finset, coe_univ, Algebra.adjoin_univ]
+#align finite_field.has_sub.sub.polynomial.is_splitting_field FiniteField.HasSub.Sub.Polynomial.isSplittingField
+
 theorem galois_poly_separable {K : Type _} [Field K] (p q : ℕ) [CharP K p] (h : p ∣ q) :
     Separable (X ^ q - X : K[X]) := by
   use 1, X ^ q - X - 1

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

@@ -37,7 +37,7 @@ noncomputable section
 
 open Polynomial
 
-open Polynomial
+open scoped Polynomial
 
 theorem galois_poly_separable {K : Type _} [Field K] (p q : ℕ) [CharP K p] (h : p ∣ q) :
     Separable (X ^ q - X : K[X]) := by

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

@@ -100,15 +100,14 @@ theorem finrank {n} (h : n ≠ 0) : FiniteDimensional.finrank (ZMod p) (GaloisFi
   simp_rw [Algebra.mem_adjoin_iff]
   intro x hx
   -- We discharge the `p = 0` separately, to avoid typeclass issues on `zmod p`.
-  cases p
+  cases p;
   cases hp
   apply Subring.closure_induction hx <;> clear! x <;> simp_rw [mem_root_set_of_ne aux]
   · rintro x (⟨r, rfl⟩ | hx)
     · simp only [aeval_X_pow, aeval_X, AlgHom.map_sub]
       rw [← map_pow, ZMod.pow_card_pow, sub_self]
     · dsimp only [GaloisField] at hx
-      rwa [mem_root_set_of_ne aux] at hx
-      infer_instance
+      rwa [mem_root_set_of_ne aux] at hx; infer_instance
   · dsimp only [g_poly]
     rw [← coeff_zero_eq_aeval_zero']
     simp only [coeff_X_pow, coeff_X_zero, sub_zero, _root_.map_eq_zero, ite_eq_right_iff,
@@ -143,7 +142,7 @@ theorem splits_zMod_x_pow_sub_x : Splits (RingHom.id (ZMod p)) (X ^ p - X) :=
     exact (ZMod.card p).symm
   have h2 := FiniteField.x_pow_card_sub_x_natDegree_eq (ZMod p) hp
   -- We discharge the `p = 0` separately, to avoid typeclass issues on `zmod p`.
-  cases p
+  cases p;
   cases hp
   rw [splits_iff_card_roots, h1, ← Finset.card_def, Finset.card_univ, h2, ZMod.card]
 #align galois_field.splits_zmod_X_pow_sub_X GaloisField.splits_zMod_x_pow_sub_x
@@ -151,10 +150,7 @@ theorem splits_zMod_x_pow_sub_x : Splits (RingHom.id (ZMod p)) (X ^ p - X) :=
 /-- A Galois field with exponent 1 is equivalent to `zmod` -/
 def equivZmodP : GaloisField p 1 ≃ₐ[ZMod p] ZMod p :=
   have h : (X ^ p ^ 1 : (ZMod p)[X]) = X ^ Fintype.card (ZMod p) := by rw [pow_one, ZMod.card p]
-  have inst : IsSplittingField (ZMod p) (ZMod p) (X ^ p ^ 1 - X) :=
-    by
-    rw [h]
-    infer_instance
+  have inst : IsSplittingField (ZMod p) (ZMod p) (X ^ p ^ 1 - X) := by rw [h]; infer_instance
   (is_splitting_field.alg_equiv (ZMod p) (X ^ p ^ 1 - X : (ZMod p)[X])).symm
 #align galois_field.equiv_zmod_p GaloisField.equivZmodP
 
@@ -198,12 +194,8 @@ variable {K : Type _} [Field K] [Fintype K] {K' : Type _} [Field K'] [Fintype K'
 def algEquivOfCardEq (p : ℕ) [Fact p.Prime] [Algebra (ZMod p) K] [Algebra (ZMod p) K']
     (hKK' : Fintype.card K = Fintype.card K') : K ≃ₐ[ZMod p] K' :=
   by
-  have : CharP K p := by
-    rw [← Algebra.charP_iff (ZMod p) K p]
-    exact ZMod.charP p
-  have : CharP K' p := by
-    rw [← Algebra.charP_iff (ZMod p) K' p]
-    exact ZMod.charP p
+  have : CharP K p := by rw [← Algebra.charP_iff (ZMod p) K p]; exact ZMod.charP p
+  have : CharP K' p := by rw [← Algebra.charP_iff (ZMod p) K' p]; exact ZMod.charP p
   choose n a hK using FiniteField.card K p
   choose n' a' hK' using FiniteField.card K' p
   rw [hK, hK'] at hKK'

mathlib commit https://github.com/leanprover-community/mathlib/commit/57e09a1296bfb4330ddf6624f1028ba186117d82

@@ -174,7 +174,7 @@ instance (priority := 100) {K K' : Type _} [Field K] [Field K'] [Finite K'] [Alg
   cases nonempty_fintype K'
   obtain ⟨p, hp⟩ := CharP.exists K
   haveI : CharP K p := hp
-  haveI : CharP K' p := charP_of_injective_algebra_map' K K' p
+  haveI : CharP K' p := charP_of_injective_algebraMap' K K' p
   exact
     IsGalois.of_separable_splitting_field
       (galois_poly_separable p (Fintype.card K')

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

@@ -40,7 +40,7 @@ open Polynomial
 open Polynomial
 
 theorem galois_poly_separable {K : Type _} [Field K] (p q : ℕ) [CharP K p] (h : p ∣ q) :
-    Separable (x ^ q - x : K[X]) := by
+    Separable (X ^ q - X : K[X]) := by
   use 1, X ^ q - X - 1
   rw [← CharP.cast_eq_zero_iff K[X] p] at h
   rw [derivative_sub, derivative_X_pow, derivative_X, C_eq_nat_cast, h]
@@ -51,7 +51,7 @@ theorem galois_poly_separable {K : Type _} [Field K] (p q : ℕ) [CharP K p] (h
 Every field with the same cardinality is (non-canonically)
 isomorphic to this field. -/
 def GaloisField (p : ℕ) [Fact p.Prime] (n : ℕ) :=
-  SplittingField (x ^ p ^ n - x : (ZMod p)[X])deriving Field
+  SplittingField (X ^ p ^ n - X : (ZMod p)[X])deriving Field
 #align galois_field GaloisField
 
 instance : Inhabited (@GaloisField 2 (Fact.mk Nat.prime_two) 1) :=
@@ -64,7 +64,7 @@ variable (p : ℕ) [Fact p.Prime] (n : ℕ)
 instance : Algebra (ZMod p) (GaloisField p n) :=
   SplittingField.algebra _
 
-instance : IsSplittingField (ZMod p) (GaloisField p n) (x ^ p ^ n - x) :=
+instance : IsSplittingField (ZMod p) (GaloisField p n) (X ^ p ^ n - X) :=
   Polynomial.IsSplittingField.splittingField _
 
 instance : CharP (GaloisField p n) p :=
@@ -134,7 +134,7 @@ theorem card (h : n ≠ 0) : Fintype.card (GaloisField p n) = p ^ n :=
   rw [Module.card_fintype b, ← FiniteDimensional.finrank_eq_card_basis b, ZMod.card, finrank p h]
 #align galois_field.card GaloisField.card
 
-theorem splits_zMod_x_pow_sub_x : Splits (RingHom.id (ZMod p)) (x ^ p - x) :=
+theorem splits_zMod_x_pow_sub_x : Splits (RingHom.id (ZMod p)) (X ^ p - X) :=
   by
   have hp : 1 < p := (Fact.out (Nat.Prime p)).one_lt
   have h1 : roots (X ^ p - X : (ZMod p)[X]) = finset.univ.val :=
@@ -150,8 +150,8 @@ theorem splits_zMod_x_pow_sub_x : Splits (RingHom.id (ZMod p)) (x ^ p - x) :=
 
 /-- A Galois field with exponent 1 is equivalent to `zmod` -/
 def equivZmodP : GaloisField p 1 ≃ₐ[ZMod p] ZMod p :=
-  have h : (x ^ p ^ 1 : (ZMod p)[X]) = x ^ Fintype.card (ZMod p) := by rw [pow_one, ZMod.card p]
-  have inst : IsSplittingField (ZMod p) (ZMod p) (x ^ p ^ 1 - x) :=
+  have h : (X ^ p ^ 1 : (ZMod p)[X]) = X ^ Fintype.card (ZMod p) := by rw [pow_one, ZMod.card p]
+  have inst : IsSplittingField (ZMod p) (ZMod p) (X ^ p ^ 1 - X) :=
     by
     rw [h]
     infer_instance
@@ -160,12 +160,12 @@ def equivZmodP : GaloisField p 1 ≃ₐ[ZMod p] ZMod p :=
 
 variable {K : Type _} [Field K] [Fintype K] [Algebra (ZMod p) K]
 
-theorem splits_x_pow_card_sub_x : Splits (algebraMap (ZMod p) K) (x ^ Fintype.card K - x) :=
+theorem splits_x_pow_card_sub_x : Splits (algebraMap (ZMod p) K) (X ^ Fintype.card K - X) :=
   (FiniteField.HasSub.Sub.Polynomial.isSplittingField K (ZMod p)).Splits
 #align galois_field.splits_X_pow_card_sub_X GaloisField.splits_x_pow_card_sub_x
 
 theorem isSplittingField_of_card_eq (h : Fintype.card K = p ^ n) :
-    IsSplittingField (ZMod p) K (x ^ p ^ n - x) :=
+    IsSplittingField (ZMod p) K (X ^ p ^ n - X) :=
   h ▸ FiniteField.HasSub.Sub.Polynomial.isSplittingField K (ZMod p)
 #align galois_field.is_splitting_field_of_card_eq GaloisField.isSplittingField_of_card_eq
 

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

Now that I am defining NNRat.cast, I want a definitive answer to this naming issue. Plenty of lemmas in mathlib already use natCast/intCast/ratCast over nat_cast/int_cast/rat_cast, and this matches with the general expectation that underscore-separated name parts correspond to a single declaration.

@@ -56,7 +56,7 @@ theorem galois_poly_separable {K : Type*} [Field K] (p q : ℕ) [CharP K p] (h :
     Separable (X ^ q - X : K[X]) := by
   use 1, X ^ q - X - 1
   rw [← CharP.cast_eq_zero_iff K[X] p] at h
-  rw [derivative_sub, derivative_X_pow, derivative_X, C_eq_nat_cast, h]
+  rw [derivative_sub, derivative_X_pow, derivative_X, C_eq_natCast, h]
   ring
 #align galois_poly_separable galois_poly_separable
 

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

@@ -121,24 +121,24 @@ theorem finrank {n} (h : n ≠ 0) : FiniteDimensional.finrank (ZMod p) (GaloisFi
   cases p; cases hp
   refine Subring.closure_induction hx ?_ ?_ ?_ ?_ ?_ ?_ <;> simp_rw [mem_rootSet_of_ne aux]
   · rintro x (⟨r, rfl⟩ | hx)
-    · simp only [map_sub, map_pow, aeval_X]
+    · simp only [g_poly, map_sub, map_pow, aeval_X]
       rw [← map_pow, ZMod.pow_card_pow, sub_self]
     · dsimp only [GaloisField] at hx
       rwa [mem_rootSet_of_ne aux] at hx
   · rw [← coeff_zero_eq_aeval_zero']
-    simp only [coeff_X_pow, coeff_X_zero, sub_zero, _root_.map_eq_zero, ite_eq_right_iff,
+    simp only [g_poly, coeff_X_pow, coeff_X_zero, sub_zero, _root_.map_eq_zero, ite_eq_right_iff,
       one_ne_zero, coeff_sub]
     intro hn
     exact Nat.not_lt_zero 1 (pow_eq_zero hn.symm ▸ hp)
-  · simp
-  · simp only [aeval_X_pow, aeval_X, AlgHom.map_sub, add_pow_char_pow, sub_eq_zero]
+  · simp [g_poly]
+  · simp only [g_poly, aeval_X_pow, aeval_X, AlgHom.map_sub, add_pow_char_pow, sub_eq_zero]
     intro x y hx hy
     rw [hx, hy]
   · intro x hx
-    simp only [sub_eq_zero, aeval_X_pow, aeval_X, AlgHom.map_sub, sub_neg_eq_add] at *
+    simp only [g_poly, sub_eq_zero, aeval_X_pow, aeval_X, AlgHom.map_sub, sub_neg_eq_add] at *
     rw [neg_pow, hx, CharP.neg_one_pow_char_pow]
     simp
-  · simp only [aeval_X_pow, aeval_X, AlgHom.map_sub, mul_pow, sub_eq_zero]
+  · simp only [g_poly, aeval_X_pow, aeval_X, AlgHom.map_sub, mul_pow, sub_eq_zero]
     intro x y hx hy
     rw [hx, hy]
 #align galois_field.finrank GaloisField.finrank

In this pull request, I have systematically eliminated the leading whitespace preceding the colon (:) within all unlabelled or unclassified porting notes. This adjustment facilitates a more efficient review process for the remaining notes by ensuring no entries are overlooked due to formatting inconsistencies.

@@ -97,7 +97,7 @@ theorem finrank {n} (h : n ≠ 0) : FiniteDimensional.finrank (ZMod p) (GaloisFi
   set g_poly := (X ^ p ^ n - X : (ZMod p)[X])
   have hp : 1 < p := h_prime.out.one_lt
   have aux : g_poly ≠ 0 := FiniteField.X_pow_card_pow_sub_X_ne_zero _ h hp
-  -- Porting note : in the statment of `key`, replaced `g_poly` by its value otherwise the
+  -- Porting note: in the statment of `key`, replaced `g_poly` by its value otherwise the
   -- proof fails
   have key : Fintype.card (g_poly.rootSet (GaloisField p n)) = g_poly.natDegree :=
     card_rootSet_eq_natDegree (galois_poly_separable p _ (dvd_pow (dvd_refl p) h))

Purely cosmetic PR

@@ -65,7 +65,7 @@ variable (p : ℕ) [Fact p.Prime] (n : ℕ)
 /-- A finite field with `p ^ n` elements.
 Every field with the same cardinality is (non-canonically)
 isomorphic to this field. -/
-def GaloisField  := SplittingField (X ^ p ^ n - X : (ZMod p)[X])
+def GaloisField := SplittingField (X ^ p ^ n - X : (ZMod p)[X])
 -- deriving Field -- Porting note: see https://github.com/leanprover-community/mathlib4/issues/5020
 #align galois_field GaloisField
 

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

@@ -48,7 +48,7 @@ instance FiniteField.isSplittingField_sub (K F : Type*) [Field K] [Fintype K]
   adjoin_rootSet' := by
     classical
     trans Algebra.adjoin F ((roots (X ^ Fintype.card K - X : K[X])).toFinset : Set K)
-    · simp only [rootSet, Polynomial.map_pow, map_X, Polynomial.map_sub]
+    · simp only [rootSet, aroots, Polynomial.map_pow, map_X, Polynomial.map_sub]
     · rw [FiniteField.roots_X_pow_card_sub_X, val_toFinset, coe_univ, Algebra.adjoin_univ]
 #align finite_field.has_sub.sub.polynomial.is_splitting_field FiniteField.isSplittingField_sub
 

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

This has nice performance benefits.

@@ -38,7 +38,7 @@ open Polynomial Finset
 
 open scoped Polynomial
 
-instance FiniteField.isSplittingField_sub (K F : Type _) [Field K] [Fintype K]
+instance FiniteField.isSplittingField_sub (K F : Type*) [Field K] [Fintype K]
     [Field F] [Algebra F K] : IsSplittingField F K (X ^ Fintype.card K - X) where
   splits' := by
     have h : (X ^ Fintype.card K - X : K[X]).natDegree = Fintype.card K :=
@@ -52,7 +52,7 @@ instance FiniteField.isSplittingField_sub (K F : Type _) [Field K] [Fintype K]
     · rw [FiniteField.roots_X_pow_card_sub_X, val_toFinset, coe_univ, Algebra.adjoin_univ]
 #align finite_field.has_sub.sub.polynomial.is_splitting_field FiniteField.isSplittingField_sub
 
-theorem galois_poly_separable {K : Type _} [Field K] (p q : ℕ) [CharP K p] (h : p ∣ q) :
+theorem galois_poly_separable {K : Type*} [Field K] (p q : ℕ) [CharP K p] (h : p ∣ q) :
     Separable (X ^ q - X : K[X]) := by
   use 1, X ^ q - X - 1
   rw [← CharP.cast_eq_zero_iff K[X] p] at h
@@ -167,7 +167,7 @@ def equivZmodP : GaloisField p 1 ≃ₐ[ZMod p] ZMod p :=
   (@IsSplittingField.algEquiv _ (ZMod p) _ _ _ (X ^ p ^ 1 - X : (ZMod p)[X]) inst).symm
 #align galois_field.equiv_zmod_p GaloisField.equivZmodP
 
-variable {K : Type _} [Field K] [Fintype K] [Algebra (ZMod p) K]
+variable {K : Type*} [Field K] [Fintype K] [Algebra (ZMod p) K]
 
 theorem splits_X_pow_card_sub_X : Splits (algebraMap (ZMod p) K) (X ^ Fintype.card K - X) :=
   (FiniteField.isSplittingField_sub K (ZMod p)).splits
@@ -179,7 +179,7 @@ theorem isSplittingField_of_card_eq (h : Fintype.card K = p ^ n) :
   h ▸ FiniteField.isSplittingField_sub K (ZMod p)
 #align galois_field.is_splitting_field_of_card_eq GaloisField.isSplittingField_of_card_eq
 
-instance (priority := 100) {K K' : Type _} [Field K] [Field K'] [Finite K'] [Algebra K K'] :
+instance (priority := 100) {K K' : Type*} [Field K] [Field K'] [Finite K'] [Algebra K K'] :
     IsGalois K K' := by
   cases nonempty_fintype K'
   obtain ⟨p, hp⟩ := CharP.exists K
@@ -200,7 +200,7 @@ end GaloisField
 
 namespace FiniteField
 
-variable {K : Type _} [Field K] [Fintype K] {K' : Type _} [Field K'] [Fintype K']
+variable {K : Type*} [Field K] [Fintype K] {K' : Type*} [Field K'] [Fintype K']
 
 /-- Uniqueness of finite fields:
   Any two finite fields of the same cardinality are (possibly non canonically) isomorphic-/

• depends on: #5966

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

@@ -2,11 +2,6 @@
 Copyright (c) 2021 Johan Commelin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Aaron Anderson, Alex J. Best, Johan Commelin, Eric Rodriguez, Ruben Van de Velde
-
-! This file was ported from Lean 3 source module field_theory.finite.galois_field
-! leanprover-community/mathlib commit 0723536a0522d24fc2f159a096fb3304bef77472
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Algebra.CharP.Algebra
 import Mathlib.Data.ZMod.Algebra
@@ -14,6 +9,8 @@ import Mathlib.FieldTheory.Finite.Basic
 import Mathlib.FieldTheory.Galois
 import Mathlib.FieldTheory.SplittingField.IsSplittingField
 
+#align_import field_theory.finite.galois_field from "leanprover-community/mathlib"@"0723536a0522d24fc2f159a096fb3304bef77472"
+
 /-!
 # Galois fields
 

Grepping for [^ .:{-] [^ :] and reviewing the results. Once I started I couldn't stop. :-)

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

@@ -152,7 +152,7 @@ theorem card (h : n ≠ 0) : Fintype.card (GaloisField p n) = p ^ n := by
 #align galois_field.card GaloisField.card
 
 theorem splits_zmod_X_pow_sub_X : Splits (RingHom.id (ZMod p)) (X ^ p - X) := by
-  have hp : 1 < p :=  h_prime.out.one_lt
+  have hp : 1 < p := h_prime.out.one_lt
   have h1 : roots (X ^ p - X : (ZMod p)[X]) = Finset.univ.val := by
     convert FiniteField.roots_X_pow_card_sub_X (ZMod p)
     exact (ZMod.card p).symm

@@ -41,7 +41,7 @@ open Polynomial Finset
 
 open scoped Polynomial
 
-instance FiniteField.HasSub.Sub.Polynomial.isSplittingField (K F : Type _) [Field K] [Fintype K]
+instance FiniteField.isSplittingField_sub (K F : Type _) [Field K] [Fintype K]
     [Field F] [Algebra F K] : IsSplittingField F K (X ^ Fintype.card K - X) where
   splits' := by
     have h : (X ^ Fintype.card K - X : K[X]).natDegree = Fintype.card K :=
@@ -53,7 +53,7 @@ instance FiniteField.HasSub.Sub.Polynomial.isSplittingField (K F : Type _) [Fiel
     trans Algebra.adjoin F ((roots (X ^ Fintype.card K - X : K[X])).toFinset : Set K)
     · simp only [rootSet, Polynomial.map_pow, map_X, Polynomial.map_sub]
     · rw [FiniteField.roots_X_pow_card_sub_X, val_toFinset, coe_univ, Algebra.adjoin_univ]
-#align finite_field.has_sub.sub.polynomial.is_splitting_field FiniteField.HasSub.Sub.Polynomial.isSplittingField
+#align finite_field.has_sub.sub.polynomial.is_splitting_field FiniteField.isSplittingField_sub
 
 theorem galois_poly_separable {K : Type _} [Field K] (p q : ℕ) [CharP K p] (h : p ∣ q) :
     Separable (X ^ q - X : K[X]) := by
@@ -72,8 +72,8 @@ def GaloisField  := SplittingField (X ^ p ^ n - X : (ZMod p)[X])
 -- deriving Field -- Porting note: see https://github.com/leanprover-community/mathlib4/issues/5020
 #align galois_field GaloisField
 
-instance : Field (GaloisField p n) := by
-  dsimp only [GaloisField]; exact SplittingField.instFieldSplittingField _
+instance : Field (GaloisField p n) :=
+  inferInstanceAs (Field (SplittingField _))
 
 instance : Inhabited (@GaloisField 2 (Fact.mk Nat.prime_two) 1) := ⟨37⟩
 
@@ -151,7 +151,7 @@ theorem card (h : n ≠ 0) : Fintype.card (GaloisField p n) = p ^ n := by
   rw [Module.card_fintype b, ← FiniteDimensional.finrank_eq_card_basis b, ZMod.card, finrank p h]
 #align galois_field.card GaloisField.card
 
-theorem splits_zMod_x_pow_sub_x : Splits (RingHom.id (ZMod p)) (X ^ p - X) := by
+theorem splits_zmod_X_pow_sub_X : Splits (RingHom.id (ZMod p)) (X ^ p - X) := by
   have hp : 1 < p :=  h_prime.out.one_lt
   have h1 : roots (X ^ p - X : (ZMod p)[X]) = Finset.univ.val := by
     convert FiniteField.roots_X_pow_card_sub_X (ZMod p)
@@ -161,7 +161,7 @@ theorem splits_zMod_x_pow_sub_x : Splits (RingHom.id (ZMod p)) (X ^ p - X) := by
   cases p; cases hp
   rw [splits_iff_card_roots, h1, ← Finset.card_def, Finset.card_univ, h2, ZMod.card]
 set_option linter.uppercaseLean3 false in
-#align galois_field.splits_zmod_X_pow_sub_X GaloisField.splits_zMod_x_pow_sub_x
+#align galois_field.splits_zmod_X_pow_sub_X GaloisField.splits_zmod_X_pow_sub_X
 
 /-- A Galois field with exponent 1 is equivalent to `ZMod` -/
 def equivZmodP : GaloisField p 1 ≃ₐ[ZMod p] ZMod p :=
@@ -172,14 +172,14 @@ def equivZmodP : GaloisField p 1 ≃ₐ[ZMod p] ZMod p :=
 
 variable {K : Type _} [Field K] [Fintype K] [Algebra (ZMod p) K]
 
-theorem splits_x_pow_card_sub_x : Splits (algebraMap (ZMod p) K) (X ^ Fintype.card K - X) :=
-  (FiniteField.HasSub.Sub.Polynomial.isSplittingField K (ZMod p)).splits
+theorem splits_X_pow_card_sub_X : Splits (algebraMap (ZMod p) K) (X ^ Fintype.card K - X) :=
+  (FiniteField.isSplittingField_sub K (ZMod p)).splits
 set_option linter.uppercaseLean3 false in
-#align galois_field.splits_X_pow_card_sub_X GaloisField.splits_x_pow_card_sub_x
+#align galois_field.splits_X_pow_card_sub_X GaloisField.splits_X_pow_card_sub_X
 
 theorem isSplittingField_of_card_eq (h : Fintype.card K = p ^ n) :
     IsSplittingField (ZMod p) K (X ^ p ^ n - X) :=
-  h ▸ FiniteField.HasSub.Sub.Polynomial.isSplittingField K (ZMod p)
+  h ▸ FiniteField.isSplittingField_sub K (ZMod p)
 #align galois_field.is_splitting_field_of_card_eq GaloisField.isSplittingField_of_card_eq
 
 instance (priority := 100) {K K' : Type _} [Field K] [Field K'] [Finite K'] [Algebra K K'] :
@@ -228,7 +228,6 @@ def ringEquivOfCardEq (hKK' : Fintype.card K = Fintype.card K') : K ≃+* K' :=
   choose n hp hK using FiniteField.card K p
   choose n' hp' hK' using FiniteField.card K' p'
   have hpp' : p = p' := by
-    -- := eq_prime_of_eq_prime_pow
     by_contra hne
     have h2 := Nat.coprime_pow_primes n n' hp hp' hne
     rw [(Eq.congr hK hK').mp hKK', Nat.coprime_self, pow_eq_one_iff (PNat.ne_zero n')] at h2

This PR should wait until the discussion here is resolved

• depends on: #5284

Co-authored-by: Riccardo Brasca <riccardo.brasca@gmail.com>