field_theory.splitting_field.is_splitting_field ⟷ Mathlib.FieldTheory.SplittingField.IsSplittingField

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

@@ -85,7 +85,7 @@ theorem splits_iff (f : K[X]) [IsSplittingField K L f] :
       adjoin_rootSet L f ▸
         Algebra.adjoin_le_iff.2 fun y hy =>
           let ⟨x, hxs, hxy⟩ :=
-            Finset.mem_image.1 (by rwa [root_set, roots_map _ h, Multiset.toFinset_map] at hy )
+            Finset.mem_image.1 (by rwa [root_set, roots_map _ h, Multiset.toFinset_map] at hy)
           hxy ▸ SetLike.mem_coe.2 <| Subalgebra.algebraMap_mem _ _,
     fun h =>
     @RingEquiv.toRingHom_refl K _ ▸
@@ -146,7 +146,7 @@ theorem finiteDimensional (f : K[X]) [IsSplittingField K L f] : FiniteDimensiona
   ⟨@Algebra.top_toSubmodule K L _ _ _ ▸
       adjoin_rootSet L f ▸
         fg_adjoin_of_finite (Finset.finite_toSet _) fun y hy =>
-          if hf : f = 0 then by rw [hf, root_set_zero] at hy ; cases hy
+          if hf : f = 0 then by rw [hf, root_set_zero] at hy; cases hy
           else
             isAlgebraic_iff_isIntegral.1
               ⟨f, hf,
@@ -181,7 +181,7 @@ variable {K L} [Field K] [Field L] [Algebra K L] {p : K[X]}
 theorem splits_of_splits {F : IntermediateField K L} (h : p.Splits (algebraMap K L))
     (hF : ∀ x ∈ p.rootSet L, x ∈ F) : p.Splits (algebraMap K F) :=
   by
-  simp_rw [root_set, Finset.mem_coe, Multiset.mem_toFinset] at hF 
+  simp_rw [root_set, Finset.mem_coe, Multiset.mem_toFinset] at hF
   rw [splits_iff_exists_multiset]
   refine' ⟨Multiset.pmap Subtype.mk _ hF, map_injective _ (algebraMap F L).Injective _⟩
   conv_lhs =>

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

@@ -47,8 +47,8 @@ namespace Polynomial
 variable [Field K] [Field L] [Field F] [Algebra K L]
 
 #print Polynomial.IsSplittingField /-
-/- ./././Mathport/Syntax/Translate/Command.lean:404:30: infer kinds are unsupported in Lean 4: #[`Splits] [] -/
-/- ./././Mathport/Syntax/Translate/Command.lean:404:30: infer kinds are unsupported in Lean 4: #[`adjoin_rootSet] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:400:30: infer kinds are unsupported in Lean 4: #[`Splits] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:400:30: infer kinds are unsupported in Lean 4: #[`adjoin_rootSet] [] -/
 /-- Typeclass characterising splitting fields. -/
 class IsSplittingField (f : K[X]) : Prop where
   Splits : Splits (algebraMap K L) f

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

@@ -47,8 +47,8 @@ namespace Polynomial
 variable [Field K] [Field L] [Field F] [Algebra K L]
 
 #print Polynomial.IsSplittingField /-
-/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`Splits] [] -/
-/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`adjoin_rootSet] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:404:30: infer kinds are unsupported in Lean 4: #[`Splits] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:404:30: infer kinds are unsupported in Lean 4: #[`adjoin_rootSet] [] -/
 /-- Typeclass characterising splitting fields. -/
 class IsSplittingField (f : K[X]) : Prop where
   Splits : Splits (algebraMap K L) f

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

@@ -145,7 +145,7 @@ def lift [Algebra K F] (f : K[X]) [IsSplittingField K L f]
 theorem finiteDimensional (f : K[X]) [IsSplittingField K L f] : FiniteDimensional K L :=
   ⟨@Algebra.top_toSubmodule K L _ _ _ ▸
       adjoin_rootSet L f ▸
-        FG_adjoin_of_finite (Finset.finite_toSet _) fun y hy =>
+        fg_adjoin_of_finite (Finset.finite_toSet _) fun y hy =>
           if hf : f = 0 then by rw [hf, root_set_zero] at hy ; cases hy
           else
             isAlgebraic_iff_isIntegral.1

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

@@ -131,7 +131,7 @@ def lift [Algebra K F] (f : K[X]) [IsSplittingField K L f]
       (by rw [← adjoin_root_set L f];
         exact
           Classical.choice
-            (lift_of_splits _ fun y hy =>
+            (Polynomial.lift_of_splits _ fun y hy =>
               have : aeval y f = 0 :=
                 (eval₂_eq_eval_map _).trans <|
                   (mem_roots <| map_ne_zero hf0).1 (multiset.mem_to_finset.mp hy)

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

@@ -3,9 +3,9 @@ Copyright (c) 2018 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes
 -/
-import Mathbin.Algebra.CharP.Algebra
-import Mathbin.FieldTheory.IntermediateField
-import Mathbin.RingTheory.Adjoin.Field
+import Algebra.CharP.Algebra
+import FieldTheory.IntermediateField
+import RingTheory.Adjoin.Field
 
 #align_import field_theory.splitting_field.is_splitting_field from "leanprover-community/mathlib"@"9fb8964792b4237dac6200193a0d533f1b3f7423"
 
@@ -47,8 +47,8 @@ namespace Polynomial
 variable [Field K] [Field L] [Field F] [Algebra K L]
 
 #print Polynomial.IsSplittingField /-
-/- ./././Mathport/Syntax/Translate/Command.lean:393:30: infer kinds are unsupported in Lean 4: #[`Splits] [] -/
-/- ./././Mathport/Syntax/Translate/Command.lean:393:30: infer kinds are unsupported in Lean 4: #[`adjoin_rootSet] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`Splits] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`adjoin_rootSet] [] -/
 /-- Typeclass characterising splitting fields. -/
 class IsSplittingField (f : K[X]) : Prop where
   Splits : Splits (algebraMap K L) f

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

@@ -2,16 +2,13 @@
 Copyright (c) 2018 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes
-
-! This file was ported from Lean 3 source module field_theory.splitting_field.is_splitting_field
-! leanprover-community/mathlib commit 9fb8964792b4237dac6200193a0d533f1b3f7423
-! 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.FieldTheory.IntermediateField
 import Mathbin.RingTheory.Adjoin.Field
 
+#align_import field_theory.splitting_field.is_splitting_field from "leanprover-community/mathlib"@"9fb8964792b4237dac6200193a0d533f1b3f7423"
+
 /-!
 # Splitting fields
 

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

@@ -50,8 +50,8 @@ namespace Polynomial
 variable [Field K] [Field L] [Field F] [Algebra K L]
 
 #print Polynomial.IsSplittingField /-
-/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`Splits] [] -/
-/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`adjoin_rootSet] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:393:30: infer kinds are unsupported in Lean 4: #[`Splits] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:393:30: infer kinds are unsupported in Lean 4: #[`adjoin_rootSet] [] -/
 /-- Typeclass characterising splitting fields. -/
 class IsSplittingField (f : K[X]) : Prop where
   Splits : Splits (algebraMap K L) f
@@ -80,6 +80,7 @@ instance map (f : F[X]) [IsSplittingField F L f] : IsSplittingField K L (f.map <
 
 variable (L)
 
+#print Polynomial.IsSplittingField.splits_iff /-
 theorem splits_iff (f : K[X]) [IsSplittingField K L f] :
     Polynomial.Splits (RingHom.id K) f ↔ (⊤ : Subalgebra K L) = ⊥ :=
   ⟨fun h =>
@@ -94,6 +95,7 @@ theorem splits_iff (f : K[X]) [IsSplittingField K L f] :
       RingEquiv.self_trans_symm (RingEquiv.ofBijective _ <| Algebra.bijective_algebraMap_iff.2 h) ▸
         by rw [RingEquiv.toRingHom_trans]; exact splits_comp_of_splits _ _ (splits L f)⟩
 #align polynomial.is_splitting_field.splits_iff Polynomial.IsSplittingField.splits_iff
+-/
 
 #print Polynomial.IsSplittingField.mul /-
 theorem mul (f g : F[X]) (hf : f ≠ 0) (hg : g ≠ 0) [IsSplittingField F K f]
@@ -178,6 +180,7 @@ open Polynomial
 
 variable {K L} [Field K] [Field L] [Algebra K L] {p : K[X]}
 
+#print IntermediateField.splits_of_splits /-
 theorem splits_of_splits {F : IntermediateField K L} (h : p.Splits (algebraMap K L))
     (hF : ∀ x ∈ p.rootSet L, x ∈ F) : p.Splits (algebraMap K F) :=
   by
@@ -191,6 +194,7 @@ theorem splits_of_splits {F : IntermediateField K L} (h : p.Splits (algebraMap K
     map_C, map_X]
   rfl
 #align intermediate_field.splits_of_splits IntermediateField.splits_of_splits
+-/
 
 end IntermediateField
 

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: Chris Hughes
 
 ! This file was ported from Lean 3 source module field_theory.splitting_field.is_splitting_field
-! leanprover-community/mathlib commit c20927220ef87bb4962ba08bf6da2ce3cf50a6dd
+! leanprover-community/mathlib commit 9fb8964792b4237dac6200193a0d533f1b3f7423
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -51,11 +51,11 @@ variable [Field K] [Field L] [Field F] [Algebra K L]
 
 #print Polynomial.IsSplittingField /-
 /- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`Splits] [] -/
-/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`adjoin_roots] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`adjoin_rootSet] [] -/
 /-- Typeclass characterising splitting fields. -/
 class IsSplittingField (f : K[X]) : Prop where
   Splits : Splits (algebraMap K L) f
-  adjoin_roots : Algebra.adjoin K (↑(f.map (algebraMap K L)).roots.toFinset : Set L) = ⊤
+  adjoin_rootSet : Algebra.adjoin K (f.rootSet L) = ⊤
 #align polynomial.is_splitting_field Polynomial.IsSplittingField
 -/
 
@@ -72,8 +72,8 @@ instance map (f : F[X]) [IsSplittingField F L f] : IsSplittingField K L (f.map <
   ⟨by rw [splits_map_iff, ← IsScalarTower.algebraMap_eq]; exact splits L f,
     Subalgebra.restrictScalars_injective F <|
       by
-      rw [map_map, ← IsScalarTower.algebraMap_eq, Subalgebra.restrictScalars_top, eq_top_iff, ←
-        adjoin_roots L f, Algebra.adjoin_le_iff]
+      rw [root_set, map_map, ← IsScalarTower.algebraMap_eq, Subalgebra.restrictScalars_top,
+        eq_top_iff, ← adjoin_root_set L f, Algebra.adjoin_le_iff]
       exact fun x hx => @Algebra.subset_adjoin K _ _ _ _ _ _ hx⟩
 #align polynomial.is_splitting_field.map Polynomial.IsSplittingField.map
 -/
@@ -84,11 +84,11 @@ theorem splits_iff (f : K[X]) [IsSplittingField K L f] :
     Polynomial.Splits (RingHom.id K) f ↔ (⊤ : Subalgebra K L) = ⊥ :=
   ⟨fun h =>
     eq_bot_iff.2 <|
-      adjoin_roots L f ▸
-        (roots_map (algebraMap K L) h).symm ▸
-          Algebra.adjoin_le_iff.2 fun y hy =>
-            let ⟨x, hxs, hxy⟩ := Finset.mem_image.1 (by rwa [Multiset.toFinset_map] at hy )
-            hxy ▸ SetLike.mem_coe.2 <| Subalgebra.algebraMap_mem _ _,
+      adjoin_rootSet L f ▸
+        Algebra.adjoin_le_iff.2 fun y hy =>
+          let ⟨x, hxs, hxy⟩ :=
+            Finset.mem_image.1 (by rwa [root_set, roots_map _ h, Multiset.toFinset_map] at hy )
+          hxy ▸ SetLike.mem_coe.2 <| Subalgebra.algebraMap_mem _ _,
     fun h =>
     @RingEquiv.toRingHom_refl K _ ▸
       RingEquiv.self_trans_symm (RingEquiv.ofBijective _ <| Algebra.bijective_algebraMap_iff.2 h) ▸
@@ -102,14 +102,14 @@ theorem mul (f g : F[X]) (hf : f ≠ 0) (hg : g ≠ 0) [IsSplittingField F K f]
       splits_mul _ (splits_comp_of_splits _ _ (splits K f))
         ((splits_map_iff _ _).1 (splits L <| g.map <| algebraMap F K)),
     by
-    rw [Polynomial.map_mul,
+    rw [root_set, Polynomial.map_mul,
       roots_mul (mul_ne_zero (map_ne_zero hf : f.map (algebraMap F L) ≠ 0) (map_ne_zero hg)),
       Multiset.toFinset_add, Finset.coe_union, Algebra.adjoin_union_eq_adjoin_adjoin,
       IsScalarTower.algebraMap_eq F K L, ← map_map,
       roots_map (algebraMap K L) ((splits_id_iff_splits <| algebraMap F K).2 <| splits K f),
-      Multiset.toFinset_map, Finset.coe_image, Algebra.adjoin_algebraMap, adjoin_roots,
-      Algebra.map_top, IsScalarTower.adjoin_range_toAlgHom, ← map_map, adjoin_roots,
-      Subalgebra.restrictScalars_top]⟩
+      Multiset.toFinset_map, Finset.coe_image, Algebra.adjoin_algebraMap, ← root_set,
+      adjoin_root_set, Algebra.map_top, IsScalarTower.adjoin_range_toAlgHom, ← map_map, ← root_set,
+      adjoin_root_set, Subalgebra.restrictScalars_top]⟩
 #align polynomial.is_splitting_field.mul Polynomial.IsSplittingField.mul
 -/
 
@@ -129,7 +129,7 @@ def lift [Algebra K F] (f : K[X]) [IsSplittingField K L f]
         exact Algebra.toTop
   else
     AlgHom.comp
-      (by rw [← adjoin_roots L f];
+      (by rw [← adjoin_root_set L f];
         exact
           Classical.choice
             (lift_of_splits _ fun y hy =>
@@ -145,9 +145,9 @@ def lift [Algebra K F] (f : K[X]) [IsSplittingField K L f]
 #print Polynomial.IsSplittingField.finiteDimensional /-
 theorem finiteDimensional (f : K[X]) [IsSplittingField K L f] : FiniteDimensional K L :=
   ⟨@Algebra.top_toSubmodule K L _ _ _ ▸
-      adjoin_roots L f ▸
+      adjoin_rootSet L f ▸
         FG_adjoin_of_finite (Finset.finite_toSet _) fun y hy =>
-          if hf : f = 0 then by rw [hf, Polynomial.map_zero, roots_zero] at hy ; cases hy
+          if hf : f = 0 then by rw [hf, root_set_zero] at hy ; cases hy
           else
             isAlgebraic_iff_isIntegral.1
               ⟨f, hf,
@@ -163,8 +163,8 @@ theorem of_algEquiv [Algebra K F] (p : K[X]) (f : F ≃ₐ[K] L) [IsSplittingFie
   · rw [← f.to_alg_hom.comp_algebra_map]
     exact splits_comp_of_splits _ _ (splits F p)
   ·
-    rw [← (Algebra.range_top_iff_surjective f.to_alg_hom).mpr f.surjective, ← root_set,
-      adjoin_root_set_eq_range (splits F p), root_set, adjoin_roots F p]
+    rw [← (Algebra.range_top_iff_surjective f.to_alg_hom).mpr f.surjective,
+      adjoin_root_set_eq_range (splits F p), adjoin_root_set F p]
 #align polynomial.is_splitting_field.of_alg_equiv Polynomial.IsSplittingField.of_algEquiv
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/31c24aa72e7b3e5ed97a8412470e904f82b81004

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes
 
 ! This file was ported from Lean 3 source module field_theory.splitting_field.is_splitting_field
-! leanprover-community/mathlib commit df76f43357840485b9d04ed5dee5ab115d420e87
+! leanprover-community/mathlib commit c20927220ef87bb4962ba08bf6da2ce3cf50a6dd
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -15,6 +15,9 @@ import Mathbin.RingTheory.Adjoin.Field
 /-!
 # Splitting fields
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 This file introduces the notion of a splitting field of a polynomial and provides an embedding from
 a splitting field to any field that splits the polynomial. A polynomial `f : K[X]` splits
 over a field extension `L` of `K` if it is zero or all of its irreducible factors over `L` have

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

@@ -46,6 +46,7 @@ namespace Polynomial
 
 variable [Field K] [Field L] [Field F] [Algebra K L]
 
+#print Polynomial.IsSplittingField /-
 /- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`Splits] [] -/
 /- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`adjoin_roots] [] -/
 /-- Typeclass characterising splitting fields. -/
@@ -53,6 +54,7 @@ class IsSplittingField (f : K[X]) : Prop where
   Splits : Splits (algebraMap K L) f
   adjoin_roots : Algebra.adjoin K (↑(f.map (algebraMap K L)).roots.toFinset : Set L) = ⊤
 #align polynomial.is_splitting_field Polynomial.IsSplittingField
+-/
 
 variable {K L F}
 
@@ -62,6 +64,7 @@ section ScalarTower
 
 variable [Algebra F K] [Algebra F L] [IsScalarTower F K L]
 
+#print Polynomial.IsSplittingField.map /-
 instance map (f : F[X]) [IsSplittingField F L f] : IsSplittingField K L (f.map <| algebraMap F K) :=
   ⟨by rw [splits_map_iff, ← IsScalarTower.algebraMap_eq]; exact splits L f,
     Subalgebra.restrictScalars_injective F <|
@@ -70,6 +73,7 @@ instance map (f : F[X]) [IsSplittingField F L f] : IsSplittingField K L (f.map <
         adjoin_roots L f, Algebra.adjoin_le_iff]
       exact fun x hx => @Algebra.subset_adjoin K _ _ _ _ _ _ hx⟩
 #align polynomial.is_splitting_field.map Polynomial.IsSplittingField.map
+-/
 
 variable (L)
 
@@ -88,6 +92,7 @@ theorem splits_iff (f : K[X]) [IsSplittingField K L f] :
         by rw [RingEquiv.toRingHom_trans]; exact splits_comp_of_splits _ _ (splits L f)⟩
 #align polynomial.is_splitting_field.splits_iff Polynomial.IsSplittingField.splits_iff
 
+#print Polynomial.IsSplittingField.mul /-
 theorem mul (f g : F[X]) (hf : f ≠ 0) (hg : g ≠ 0) [IsSplittingField F K f]
     [IsSplittingField K L (g.map <| algebraMap F K)] : IsSplittingField F L (f * g) :=
   ⟨(IsScalarTower.algebraMap_eq F K L).symm ▸
@@ -103,11 +108,13 @@ theorem mul (f g : F[X]) (hf : f ≠ 0) (hg : g ≠ 0) [IsSplittingField F K f]
       Algebra.map_top, IsScalarTower.adjoin_range_toAlgHom, ← map_map, adjoin_roots,
       Subalgebra.restrictScalars_top]⟩
 #align polynomial.is_splitting_field.mul Polynomial.IsSplittingField.mul
+-/
 
 end ScalarTower
 
 variable (L)
 
+#print Polynomial.IsSplittingField.lift /-
 /-- Splitting field of `f` embeds into any field that splits `f`. -/
 def lift [Algebra K F] (f : K[X]) [IsSplittingField K L f]
     (hf : Polynomial.Splits (algebraMap K F) f) : L →ₐ[K] F :=
@@ -130,7 +137,9 @@ def lift [Algebra K F] (f : K[X]) [IsSplittingField K L f]
                 splits_of_splits_of_dvd _ hf0 hf <| minpoly.dvd _ _ this⟩))
       Algebra.toTop
 #align polynomial.is_splitting_field.lift Polynomial.IsSplittingField.lift
+-/
 
+#print Polynomial.IsSplittingField.finiteDimensional /-
 theorem finiteDimensional (f : K[X]) [IsSplittingField K L f] : FiniteDimensional K L :=
   ⟨@Algebra.top_toSubmodule K L _ _ _ ▸
       adjoin_roots L f ▸
@@ -142,7 +151,9 @@ theorem finiteDimensional (f : K[X]) [IsSplittingField K L f] : FiniteDimensiona
                 (eval₂_eq_eval_map _).trans <|
                   (mem_roots <| map_ne_zero hf).1 (Multiset.mem_toFinset.mp hy)⟩⟩
 #align polynomial.is_splitting_field.finite_dimensional Polynomial.IsSplittingField.finiteDimensional
+-/
 
+#print Polynomial.IsSplittingField.of_algEquiv /-
 theorem of_algEquiv [Algebra K F] (p : K[X]) (f : F ≃ₐ[K] L) [IsSplittingField K F p] :
     IsSplittingField K L p := by
   constructor
@@ -152,6 +163,7 @@ theorem of_algEquiv [Algebra K F] (p : K[X]) (f : F ≃ₐ[K] L) [IsSplittingFie
     rw [← (Algebra.range_top_iff_surjective f.to_alg_hom).mpr f.surjective, ← root_set,
       adjoin_root_set_eq_range (splits F p), root_set, adjoin_roots F p]
 #align polynomial.is_splitting_field.of_alg_equiv Polynomial.IsSplittingField.of_algEquiv
+-/
 
 end IsSplittingField
 

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

This uses the improved shake script from #9772 to reduce imports across mathlib. The corresponding noshake.json file has been added to #9772.

Co-authored-by: Mario Carneiro <di.gama@gmail.com>

@@ -3,7 +3,6 @@ Copyright (c) 2018 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes
 -/
-import Mathlib.Algebra.CharP.Algebra
 import Mathlib.FieldTheory.IntermediateField
 import Mathlib.RingTheory.Adjoin.Field
 

• generalize image_rootSet, adjoin_rootSet_eq_range and splits_comp_of_splits in Data/Polynomial/Splits and use the last one to golf splits_of_algHom, splits_of_isScalarTower (introduced in # 8609).

• add three new lemmas mem_range_x_of_minpoly_splits to simplify the construction of IntermediateField.algHomEquivAlgHomOfIsAlgClosed and Algebra.IsAlgebraic.algHomEquivAlgHomOfIsAlgClosed, remove the IsAlgClosed condition and rename. They could be moved to an earlier file but I refrain from doing that. (#find_home says it's already in the right place)

• golf primitive_element_iff_algHom_eq_of_eval from # 8609, using a new lemma IsIntegral.minpoly_splits_tower_top for the last step.

• make integralClosure_algEquiv_restrict (from # 8714) computable and rename to AlgEquiv.mapIntegralClosure to follow camelCase naming convention and enable dot notation.

Co-authored-by: Xavier-François Roblot <46200072+xroblot@users.noreply.github.com> Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>

@@ -151,16 +151,16 @@ end IsSplittingField
 
 end Polynomial
 
-namespace IntermediateField
-
 open Polynomial
 
-variable {K L} [Field K] [Field L] [Algebra K L] {p : K[X]}
+variable {K L} [Field K] [Field L] [Algebra K L] {p : K[X]} {F : IntermediateField K L}
 
-theorem splits_of_splits {F : IntermediateField K L} (h : p.Splits (algebraMap K L))
+theorem IntermediateField.splits_of_splits (h : p.Splits (algebraMap K L))
     (hF : ∀ x ∈ p.rootSet L, x ∈ F) : p.Splits (algebraMap K F) := by
   simp_rw [← F.fieldRange_val, rootSet_def, Finset.mem_coe, Multiset.mem_toFinset] at hF
   exact splits_of_comp _ F.val.toRingHom h hF
 #align intermediate_field.splits_of_splits IntermediateField.splits_of_splits
 
-end IntermediateField
+theorem IsIntegral.mem_intermediateField_of_minpoly_splits {x : L} (int : IsIntegral K x)
+    {F : IntermediateField K L} (h : Splits (algebraMap K F) (minpoly K x)) : x ∈ F := by
+  rw [← F.fieldRange_val]; exact int.mem_range_algebraMap_of_minpoly_splits h

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>

@@ -123,7 +123,7 @@ def lift [Algebra K F] (f : K[X]) [IsSplittingField K L f]
     exact Classical.choice (lift_of_splits _ fun y hy =>
       have : aeval y f = 0 := (eval₂_eq_eval_map _).trans <|
         (mem_roots <| map_ne_zero hf0).1 (Multiset.mem_toFinset.mp hy)
-    ⟨isAlgebraic_iff_isIntegral.1 ⟨f, hf0, this⟩,
+    ⟨IsAlgebraic.isIntegral ⟨f, hf0, this⟩,
       splits_of_splits_of_dvd _ hf0 hf <| minpoly.dvd _ _ this⟩)) Algebra.toTop
 #align polynomial.is_splitting_field.lift Polynomial.IsSplittingField.lift
 
@@ -131,7 +131,7 @@ theorem finiteDimensional (f : K[X]) [IsSplittingField K L f] : FiniteDimensiona
   ⟨@Algebra.top_toSubmodule K L _ _ _ ▸
     adjoin_rootSet L f ▸ fg_adjoin_of_finite (Finset.finite_toSet _) fun y hy ↦
       if hf : f = 0 then by rw [hf, rootSet_zero] at hy; cases hy
-      else isAlgebraic_iff_isIntegral.1 ⟨f, hf, (mem_rootSet'.mp hy).2⟩⟩
+      else IsAlgebraic.isIntegral ⟨f, hf, (mem_rootSet'.mp hy).2⟩⟩
 #align polynomial.is_splitting_field.finite_dimensional Polynomial.IsSplittingField.finiteDimensional
 
 theorem of_algEquiv [Algebra K F] (p : K[X]) (f : F ≃ₐ[K] L) [IsSplittingField K F p] :

This PR tests a string-based tool for renaming declarations.

Inspired by this Zulip thread, I am trying to reduce the diff of #8406.

This PR makes the following renames:

| From | To |

@@ -129,7 +129,7 @@ def lift [Algebra K F] (f : K[X]) [IsSplittingField K L f]
 
 theorem finiteDimensional (f : K[X]) [IsSplittingField K L f] : FiniteDimensional K L :=
   ⟨@Algebra.top_toSubmodule K L _ _ _ ▸
-    adjoin_rootSet L f ▸ FG_adjoin_of_finite (Finset.finite_toSet _) fun y hy ↦
+    adjoin_rootSet L f ▸ fg_adjoin_of_finite (Finset.finite_toSet _) fun y hy ↦
       if hf : f = 0 then by rw [hf, rootSet_zero] at hy; cases hy
       else isAlgebraic_iff_isIntegral.1 ⟨f, hf, (mem_rootSet'.mp hy).2⟩⟩
 #align polynomial.is_splitting_field.finite_dimensional Polynomial.IsSplittingField.finiteDimensional

Before: Construction only imports Normal, which transitively imports IsSplittingField Now: Normal imports Construction, Construction only imports IsSplittingField So no extra transitive import is added to any file other than Construction and Normal. As a consequence, Polynomial.SplittingField.instNormal is moved from Construction to Normal.

adjoin_rootSet_eq_range is added to IsSplittingField.

splits_of_comp in Splits is extracted from splits_of_splits in IsSplittingField.

Source of proof: https://math.stackexchange.com/a/2585087/12932

Move Algebra.adjoin.liftSingleton from IsAlgClosed/Basic to Adjoin/Field in order to speed up lift_of_splits (renamed to add namespace Polynomial).

Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>

@@ -79,7 +79,7 @@ instance map (f : F[X]) [IsSplittingField F L f] : IsSplittingField K L (f.map <
 #align polynomial.is_splitting_field.map Polynomial.IsSplittingField.map
 
 theorem splits_iff (f : K[X]) [IsSplittingField K L f] :
-    Polynomial.Splits (RingHom.id K) f ↔ (⊤ : Subalgebra K L) = ⊥ :=
+    Splits (RingHom.id K) f ↔ (⊤ : Subalgebra K L) = ⊥ :=
   ⟨fun h => by -- Porting note: replaced term-mode proof
     rw [eq_bot_iff, ← adjoin_rootSet L f, rootSet, aroots, roots_map (algebraMap K L) h,
       Algebra.adjoin_le_iff]
@@ -112,7 +112,7 @@ end ScalarTower
 
 /-- Splitting field of `f` embeds into any field that splits `f`. -/
 def lift [Algebra K F] (f : K[X]) [IsSplittingField K L f]
-    (hf : Polynomial.Splits (algebraMap K F) f) : L →ₐ[K] F :=
+    (hf : Splits (algebraMap K F) f) : L →ₐ[K] F :=
   if hf0 : f = 0 then
     (Algebra.ofId K F).comp <|
       (Algebra.botEquiv K L : (⊥ : Subalgebra K L) →ₐ[K] K).comp <| by
@@ -129,11 +129,9 @@ def lift [Algebra K F] (f : K[X]) [IsSplittingField K L f]
 
 theorem finiteDimensional (f : K[X]) [IsSplittingField K L f] : FiniteDimensional K L :=
   ⟨@Algebra.top_toSubmodule K L _ _ _ ▸
-    adjoin_rootSet L f ▸ FG_adjoin_of_finite (Finset.finite_toSet _) fun y hy =>
+    adjoin_rootSet L f ▸ FG_adjoin_of_finite (Finset.finite_toSet _) fun y hy ↦
       if hf : f = 0 then by rw [hf, rootSet_zero] at hy; cases hy
-      else
-        isAlgebraic_iff_isIntegral.1 ⟨f, hf, (eval₂_eq_eval_map _).trans <|
-          (mem_roots <| map_ne_zero hf).1 (Multiset.mem_toFinset.mp hy)⟩⟩
+      else isAlgebraic_iff_isIntegral.1 ⟨f, hf, (mem_rootSet'.mp hy).2⟩⟩
 #align polynomial.is_splitting_field.finite_dimensional Polynomial.IsSplittingField.finiteDimensional
 
 theorem of_algEquiv [Algebra K F] (p : K[X]) (f : F ≃ₐ[K] L) [IsSplittingField K F p] :
@@ -145,6 +143,10 @@ theorem of_algEquiv [Algebra K F] (p : K[X]) (f : F ≃ₐ[K] L) [IsSplittingFie
       adjoin_rootSet_eq_range (splits F p), adjoin_rootSet F p]
 #align polynomial.is_splitting_field.of_alg_equiv Polynomial.IsSplittingField.of_algEquiv
 
+theorem adjoin_rootSet_eq_range [Algebra K F] (f : K[X]) [IsSplittingField K L f] (i : L →ₐ[K] F) :
+    Algebra.adjoin K (rootSet f F) = i.range :=
+  (Polynomial.adjoin_rootSet_eq_range (splits L f) i).mpr (adjoin_rootSet L f)
+
 end IsSplittingField
 
 end Polynomial
@@ -157,15 +159,8 @@ variable {K L} [Field K] [Field L] [Algebra K L] {p : K[X]}
 
 theorem splits_of_splits {F : IntermediateField K L} (h : p.Splits (algebraMap K L))
     (hF : ∀ x ∈ p.rootSet L, x ∈ F) : p.Splits (algebraMap K F) := by
-  simp_rw [rootSet_def, Finset.mem_coe, Multiset.mem_toFinset] at hF
-  rw [splits_iff_exists_multiset]
-  refine' ⟨Multiset.pmap Subtype.mk _ hF, map_injective _ (algebraMap F L).injective _⟩
-  conv_lhs =>
-    rw [Polynomial.map_map, ← IsScalarTower.algebraMap_eq, eq_prod_roots_of_splits h, ←
-      Multiset.pmap_eq_map _ _ _ hF]
-  simp_rw [Polynomial.map_mul, Polynomial.map_multiset_prod, Multiset.map_pmap, Polynomial.map_sub,
-    map_C, map_X]
-  rfl
+  simp_rw [← F.fieldRange_val, rootSet_def, Finset.mem_coe, Multiset.mem_toFinset] at hF
+  exact splits_of_comp _ F.val.toRingHom h hF
 #align intermediate_field.splits_of_splits IntermediateField.splits_of_splits
 
 end IntermediateField

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

@@ -73,7 +73,7 @@ variable [Algebra F K] [Algebra F L] [IsScalarTower F K L]
 instance map (f : F[X]) [IsSplittingField F L f] : IsSplittingField K L (f.map <| algebraMap F K) :=
   ⟨by rw [splits_map_iff, ← IsScalarTower.algebraMap_eq]; exact splits L f,
     Subalgebra.restrictScalars_injective F <| by
-      rw [rootSet, map_map, ← IsScalarTower.algebraMap_eq, Subalgebra.restrictScalars_top,
+      rw [rootSet, aroots, map_map, ← IsScalarTower.algebraMap_eq, Subalgebra.restrictScalars_top,
         eq_top_iff, ← adjoin_rootSet L f, Algebra.adjoin_le_iff]
       exact fun x hx => @Algebra.subset_adjoin K _ _ _ _ _ _ hx⟩
 #align polynomial.is_splitting_field.map Polynomial.IsSplittingField.map
@@ -81,7 +81,7 @@ instance map (f : F[X]) [IsSplittingField F L f] : IsSplittingField K L (f.map <
 theorem splits_iff (f : K[X]) [IsSplittingField K L f] :
     Polynomial.Splits (RingHom.id K) f ↔ (⊤ : Subalgebra K L) = ⊥ :=
   ⟨fun h => by -- Porting note: replaced term-mode proof
-    rw [eq_bot_iff, ← adjoin_rootSet L f, rootSet, roots_map (algebraMap K L) h,
+    rw [eq_bot_iff, ← adjoin_rootSet L f, rootSet, aroots, roots_map (algebraMap K L) h,
       Algebra.adjoin_le_iff]
     intro y hy
     rw [Multiset.toFinset_map, Finset.mem_coe, Finset.mem_image] at hy
@@ -99,10 +99,9 @@ theorem mul (f g : F[X]) (hf : f ≠ 0) (hg : g ≠ 0) [IsSplittingField F K f]
   ⟨(IsScalarTower.algebraMap_eq F K L).symm ▸
       splits_mul _ (splits_comp_of_splits _ _ (splits K f))
         ((splits_map_iff _ _).1 (splits L <| g.map <| algebraMap F K)), by
-    rw [rootSet, Polynomial.map_mul,
-      roots_mul (mul_ne_zero (map_ne_zero hf : f.map (algebraMap F L) ≠ 0) (map_ne_zero hg)),
+    rw [rootSet, aroots_mul (mul_ne_zero hf hg),
       Multiset.toFinset_add, Finset.coe_union, Algebra.adjoin_union_eq_adjoin_adjoin,
-      IsScalarTower.algebraMap_eq F K L, ← map_map,
+      aroots_def, aroots_def, IsScalarTower.algebraMap_eq F K L, ← map_map,
       roots_map (algebraMap K L) ((splits_id_iff_splits <| algebraMap F K).2 <| splits K f),
       Multiset.toFinset_map, Finset.coe_image, Algebra.adjoin_algebraMap, ← rootSet, adjoin_rootSet,
       Algebra.map_top, IsScalarTower.adjoin_range_toAlgHom, ← map_map, ← rootSet, adjoin_rootSet,

• 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) 2018 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes
-
-! This file was ported from Lean 3 source module field_theory.splitting_field.is_splitting_field
-! 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.Algebra.CharP.Algebra
 import Mathlib.FieldTheory.IntermediateField
 import Mathlib.RingTheory.Adjoin.Field
 
+#align_import field_theory.splitting_field.is_splitting_field from "leanprover-community/mathlib"@"9fb8964792b4237dac6200193a0d533f1b3f7423"
+
 /-!
 # Splitting fields
 

The important thing to forward-port here is the addition of [DecidableEq _] to a handful of lemmas.

linear_algebra.eigenspace.minpoly did not need anything forward-porting, as the proof which broke in mathlib3 did not break in mathlib4.

The nthRootsFinset_def lemma was forgotten in the mathlib3 PR.

@@ -161,7 +161,7 @@ variable {K L} [Field K] [Field L] [Algebra K L] {p : K[X]}
 
 theorem splits_of_splits {F : IntermediateField K L} (h : p.Splits (algebraMap K L))
     (hF : ∀ x ∈ p.rootSet L, x ∈ F) : p.Splits (algebraMap K F) := by
-  simp_rw [rootSet, Finset.mem_coe, Multiset.mem_toFinset] at hF
+  simp_rw [rootSet_def, Finset.mem_coe, Multiset.mem_toFinset] at hF
   rw [splits_iff_exists_multiset]
   refine' ⟨Multiset.pmap Subtype.mk _ hF, map_injective _ (algebraMap F L).injective _⟩
   conv_lhs =>

Strangely, making one proof use fewer simp lemmas has made the proof slower (and the hearbeats have been bumped accordingly)

Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au>

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes
 
 ! This file was ported from Lean 3 source module field_theory.splitting_field.is_splitting_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.
 -/
@@ -49,7 +49,7 @@ variable [Field K] [Field L] [Field F] [Algebra K L]
 /-- Typeclass characterising splitting fields. -/
 class IsSplittingField (f : K[X]) : Prop where
   splits' : Splits (algebraMap K L) f
-  adjoin_roots' : Algebra.adjoin K (↑(f.map (algebraMap K L)).roots.toFinset : Set L) = ⊤
+  adjoin_rootSet' : Algebra.adjoin K (f.rootSet L : Set L) = ⊤
 #align polynomial.is_splitting_field Polynomial.IsSplittingField
 
 namespace IsSplittingField
@@ -63,11 +63,11 @@ theorem splits (f : K[X]) [IsSplittingField K L f] : Splits (algebraMap K L) f :
 #align polynomial.is_splitting_field.splits Polynomial.IsSplittingField.splits
 
 -- Porting note: infer kinds are unsupported
--- so we provide a version of `adjoin_roots'` with `f` explicit.
-theorem adjoin_roots (f : K[X]) [IsSplittingField K L f] :
-    Algebra.adjoin K (↑(f.map (algebraMap K L)).roots.toFinset : Set L) = ⊤ :=
-  adjoin_roots'
-#align polynomial.is_splitting_field.adjoin_roots Polynomial.IsSplittingField.adjoin_roots
+-- so we provide a version of `adjoin_rootSet'` with `f` explicit.
+theorem adjoin_rootSet (f : K[X]) [IsSplittingField K L f] :
+    Algebra.adjoin K (f.rootSet L : Set L) = ⊤ :=
+  adjoin_rootSet'
+#align polynomial.is_splitting_field.adjoin_root_set Polynomial.IsSplittingField.adjoin_rootSet
 
 section ScalarTower
 
@@ -76,15 +76,16 @@ variable [Algebra F K] [Algebra F L] [IsScalarTower F K L]
 instance map (f : F[X]) [IsSplittingField F L f] : IsSplittingField K L (f.map <| algebraMap F K) :=
   ⟨by rw [splits_map_iff, ← IsScalarTower.algebraMap_eq]; exact splits L f,
     Subalgebra.restrictScalars_injective F <| by
-      rw [map_map, ← IsScalarTower.algebraMap_eq, Subalgebra.restrictScalars_top, eq_top_iff, ←
-        adjoin_roots L f, Algebra.adjoin_le_iff]
+      rw [rootSet, map_map, ← IsScalarTower.algebraMap_eq, Subalgebra.restrictScalars_top,
+        eq_top_iff, ← adjoin_rootSet L f, Algebra.adjoin_le_iff]
       exact fun x hx => @Algebra.subset_adjoin K _ _ _ _ _ _ hx⟩
 #align polynomial.is_splitting_field.map Polynomial.IsSplittingField.map
 
 theorem splits_iff (f : K[X]) [IsSplittingField K L f] :
     Polynomial.Splits (RingHom.id K) f ↔ (⊤ : Subalgebra K L) = ⊥ :=
   ⟨fun h => by -- Porting note: replaced term-mode proof
-    rw [eq_bot_iff, ← adjoin_roots L f, roots_map (algebraMap K L) h, Algebra.adjoin_le_iff]
+    rw [eq_bot_iff, ← adjoin_rootSet L f, rootSet, roots_map (algebraMap K L) h,
+      Algebra.adjoin_le_iff]
     intro y hy
     rw [Multiset.toFinset_map, Finset.mem_coe, Finset.mem_image] at hy
     obtain ⟨x : K, -, hxy : algebraMap K L x = y⟩ := hy
@@ -101,13 +102,13 @@ theorem mul (f g : F[X]) (hf : f ≠ 0) (hg : g ≠ 0) [IsSplittingField F K f]
   ⟨(IsScalarTower.algebraMap_eq F K L).symm ▸
       splits_mul _ (splits_comp_of_splits _ _ (splits K f))
         ((splits_map_iff _ _).1 (splits L <| g.map <| algebraMap F K)), by
-    rw [Polynomial.map_mul,
+    rw [rootSet, Polynomial.map_mul,
       roots_mul (mul_ne_zero (map_ne_zero hf : f.map (algebraMap F L) ≠ 0) (map_ne_zero hg)),
       Multiset.toFinset_add, Finset.coe_union, Algebra.adjoin_union_eq_adjoin_adjoin,
       IsScalarTower.algebraMap_eq F K L, ← map_map,
       roots_map (algebraMap K L) ((splits_id_iff_splits <| algebraMap F K).2 <| splits K f),
-      Multiset.toFinset_map, Finset.coe_image, Algebra.adjoin_algebraMap, adjoin_roots,
-      Algebra.map_top, IsScalarTower.adjoin_range_toAlgHom, ← map_map, adjoin_roots,
+      Multiset.toFinset_map, Finset.coe_image, Algebra.adjoin_algebraMap, ← rootSet, adjoin_rootSet,
+      Algebra.map_top, IsScalarTower.adjoin_range_toAlgHom, ← map_map, ← rootSet, adjoin_rootSet,
       Subalgebra.restrictScalars_top]⟩
 #align polynomial.is_splitting_field.mul Polynomial.IsSplittingField.mul
 
@@ -122,7 +123,7 @@ def lift [Algebra K F] (f : K[X]) [IsSplittingField K L f]
         rw [← (splits_iff L f).1 (show f.Splits (RingHom.id K) from hf0.symm ▸ splits_zero _)]
         exact Algebra.toTop
   else AlgHom.comp (by
-    rw [← adjoin_roots L f];
+    rw [← adjoin_rootSet L f];
     exact Classical.choice (lift_of_splits _ fun y hy =>
       have : aeval y f = 0 := (eval₂_eq_eval_map _).trans <|
         (mem_roots <| map_ne_zero hf0).1 (Multiset.mem_toFinset.mp hy)
@@ -132,8 +133,9 @@ def lift [Algebra K F] (f : K[X]) [IsSplittingField K L f]
 
 theorem finiteDimensional (f : K[X]) [IsSplittingField K L f] : FiniteDimensional K L :=
   ⟨@Algebra.top_toSubmodule K L _ _ _ ▸
-    adjoin_roots L f ▸ FG_adjoin_of_finite (Finset.finite_toSet _) fun y hy =>
-      if hf : f = 0 then by rw [hf, Polynomial.map_zero, roots_zero] at hy; cases hy else
+    adjoin_rootSet L f ▸ FG_adjoin_of_finite (Finset.finite_toSet _) fun y hy =>
+      if hf : f = 0 then by rw [hf, rootSet_zero] at hy; cases hy
+      else
         isAlgebraic_iff_isIntegral.1 ⟨f, hf, (eval₂_eq_eval_map _).trans <|
           (mem_roots <| map_ne_zero hf).1 (Multiset.mem_toFinset.mp hy)⟩⟩
 #align polynomial.is_splitting_field.finite_dimensional Polynomial.IsSplittingField.finiteDimensional
@@ -143,8 +145,8 @@ theorem of_algEquiv [Algebra K F] (p : K[X]) (f : F ≃ₐ[K] L) [IsSplittingFie
   constructor
   · rw [← f.toAlgHom.comp_algebraMap]
     exact splits_comp_of_splits _ _ (splits F p)
-  · rw [← (Algebra.range_top_iff_surjective f.toAlgHom).mpr f.surjective, ← rootSet,
-      adjoin_rootSet_eq_range (splits F p), rootSet, adjoin_roots F p]
+  · rw [← (Algebra.range_top_iff_surjective f.toAlgHom).mpr f.surjective,
+      adjoin_rootSet_eq_range (splits F p), adjoin_rootSet F p]
 #align polynomial.is_splitting_field.of_alg_equiv Polynomial.IsSplittingField.of_algEquiv
 
 end IsSplittingField

Co-authored-by: Johan Commelin <johan@commelin.net>