feat(FieldTheory/KrullTopology): define uniform group structure on galois group

Mathlib/FieldTheory/AbsoluteGaloisGroup.lean

@@ -52,7 +52,7 @@ section
 
 variable [Algebra K L] [Algebra (AlgebraicClosure K) (AlgebraicClosure L)]
   [IsScalarTower K (AlgebraicClosure K) (AlgebraicClosure L)]
-
+-- set_option pp.rawOnError true
 open IntermediateField in
 /-- A commuting square of two fields and their algebraic closures induces a continuous homomorphism
 of their absolute Galois groups. -/
@@ -65,10 +65,11 @@ noncomputable def absoluteGaloisGroup.mapOfAlgebra : G_K L →ₜ* G_K K :=
       let f := IsScalarTower.toAlgHom K (AlgebraicClosure K) (AlgebraicClosure L)
       apply continuous_of_continuousAt_one F
       rw [ContinuousAt, map_one]
-      refine ((galGroupBasis L (AlgebraicClosure L)).nhds_one_hasBasis.tendsto_iff
-        (galGroupBasis K (AlgebraicClosure K)).nhds_one_hasBasis).mpr ?_
-      rintro _ ⟨_, ⟨F, hF : FiniteDimensional _ _, rfl⟩, rfl⟩
-      refine ⟨_, ⟨_, ⟨adjoin L (F.map f), ?_, rfl⟩, rfl⟩, fun σ hσ x ↦ ?_⟩
+      intro U hU
+      rw [Filter.mem_map]
+      erw [krullTopology_mem_nhds_one_iff] at hU ⊢
+      obtain ⟨F, fd, hF⟩ := hU
+      refine ⟨adjoin L (F.map f), ?_, fun σ hσ => hF fun x => ?_⟩
       · suffices Algebra.EssFiniteType L (adjoin L (F.map f : Set (AlgebraicClosure L))) by
           apply Algebra.finite_of_essFiniteType_of_isAlgebraic
         replace hF : Algebra.EssFiniteType K F := inferInstance

Mathlib/FieldTheory/Galois/Basic.lean

@@ -270,6 +270,16 @@ def fixingSubgroupEquiv : fixingSubgroup K ≃* Gal(E/K) where
   invFun ϕ := ⟨ϕ.restrictScalars _, ϕ.commutes⟩
   map_mul' _ _ := by ext; rfl
 
+@[simp]
+theorem coe_fixingSubgroupEquiv_apply (σ : K.fixingSubgroup) :
+    ⇑(K.fixingSubgroupEquiv σ) = ⇑σ.1 :=
+  rfl
+
+@[simp]
+theorem coe_fixingSubgroupEquiv_symm_apply (σ : Gal(E/K)) :
+    ⇑(K.fixingSubgroupEquiv.symm σ).1 = ⇑σ :=
+  rfl
+
 theorem fixingSubgroup_fixedField [FiniteDimensional F E] : fixingSubgroup (fixedField H) = H := by
   have H_le : H ≤ fixingSubgroup (fixedField H) := (le_iff_le _ _).mp le_rfl
   suffices Nat.card H = Nat.card (fixingSubgroup (fixedField H)) by
@@ -288,6 +298,16 @@ def subgroupEquivAlgEquiv [FiniteDimensional F E] (H : Subgroup Gal(E/F)) :
     H ≃* Gal(E/IntermediateField.fixedField H) :=
   (MulEquiv.subgroupCongr (fixingSubgroup_fixedField H).symm).trans (fixingSubgroupEquiv _)
 
+@[simp]
+theorem coe_subgroupEquivAlgEquiv_apply [FiniteDimensional F E] (σ : H) :
+    ⇑(subgroupEquivAlgEquiv H σ) = ⇑σ.1 :=
+  rfl
+
+@[simp]
+theorem coe_subgroupEquivAlgEquiv_symm_apply [FiniteDimensional F E] (σ : Gal(E/fixedField H)) :
+    ⇑((subgroupEquivAlgEquiv H).symm σ).1 = ⇑σ :=
+  rfl
+
 instance fixedField.smul : SMul K (fixedField (fixingSubgroup K)) where
   smul x y := ⟨x * y, fun ϕ => by
     rw [smul_mul', show ϕ • (x : E) = ↑x from ϕ.2 x, show ϕ • (y : E) = ↑y from y.2 ϕ]⟩

Mathlib/FieldTheory/Galois/Infinite.lean

@@ -22,8 +22,6 @@ closed subgroups of the Galois group.
 
 In `K/k`, for any intermediate field `L` :
 
-* `fixingSubgroup_isClosed` : the subgroup fixing `L` (`Gal(K/L)`) is closed.
-
 * `fixedField_fixingSubgroup` : the field fixed by the
   subgroup fixing `L` is equal to `L` itself.
 
@@ -61,26 +59,10 @@ open scoped Pointwise
 open FiniteGaloisIntermediateField AlgEquiv
 --Note: The `adjoin`s below are `FiniteGaloisIntermediateField.adjoin`
 
+@[deprecated IntermediateField.isClosed_fixingSubgroup (since := "2026-05-14")]
 lemma fixingSubgroup_isClosed (L : IntermediateField k K) [IsGalois k K] :
-    IsClosed (L.fixingSubgroup : Set Gal(K/k)) where
-  isOpen_compl := isOpen_iff_mem_nhds.mpr fun σ h => by
-    apply mem_nhds_iff.mpr
-    rcases Set.not_subset.mp ((mem_fixingSubgroup_iff Gal(K/k)).not.mp h) with ⟨y, yL, ne⟩
-    use σ • ((adjoin k {y}).1.fixingSubgroup : Set Gal(K/k))
-    constructor
-    · intro f hf
-      rcases (Set.mem_smul_set.mp hf) with ⟨g, hg, eq⟩
-      simp only [Set.mem_compl_iff, SetLike.mem_coe, ← eq]
-      apply (mem_fixingSubgroup_iff Gal(K/k)).not.mpr
-      push Not
-      use y
-      simp only [yL, smul_eq_mul, AlgEquiv.smul_def, AlgEquiv.mul_apply, ne_eq, true_and]
-      have : g y = y := (mem_fixingSubgroup_iff Gal(K/k)).mp hg y <|
-        adjoin_simple_le_iff.mp le_rfl
-      simpa only [this, ne_eq, AlgEquiv.smul_def] using! ne
-    · simp only [(IntermediateField.fixingSubgroup_isOpen (adjoin k {y}).1).smul σ, true_and]
-      use 1
-      simp only [SetLike.mem_coe, smul_eq_mul, mul_one, and_true, Subgroup.one_mem]
+    IsClosed (L.fixingSubgroup : Set Gal(K/k)) :=
+  IntermediateField.isClosed_fixingSubgroup L
 
 lemma fixedField_fixingSubgroup (L : IntermediateField k K) [IsGalois k K] :
     IntermediateField.fixedField L.fixingSubgroup = L := by
@@ -150,21 +132,17 @@ lemma fixingSubgroup_fixedField (H : ClosedSubgroup Gal(K/k)) [IsGalois k K] :
   intro σ hσ
   by_contra h
   have nhds : H.carrierᶜ ∈ nhds σ := H.isClosed'.isOpen_compl.mem_nhds h
-  rw [GroupFilterBasis.nhds_eq (x₀ := σ) (galGroupBasis k K)] at nhds
-  rcases nhds with ⟨b, ⟨gp, ⟨L, hL, eq'⟩, eq⟩, sub⟩
-  rw [← eq'] at eq
-  have := hL.out
+  rw [← map_mul_left_nhds_one, Filter.mem_map, krullTopology_mem_nhds_one_iff] at nhds
+  rcases nhds with ⟨L, hL, sub⟩
   let L' : FiniteGaloisIntermediateField k K := {
     normalClosure k L K with
     finiteDimensional := normalClosure.is_finiteDimensional k L K
     isGalois := IsGalois.normalClosure k L K }
-  have compl : σ • L'.1.fixingSubgroup.carrier ⊆ H.carrierᶜ := by
-    rintro φ ⟨τ, hτ, muleq⟩
-    have sub' : σ • b ⊆ H.carrierᶜ := Set.smul_set_subset_iff.mpr sub
-    apply sub'
-    simp only [← muleq, ← eq]
-    apply Set.smul_mem_smul_set
-    exact (L.fixingSubgroup_le (IntermediateField.le_normalClosure L) hτ)
+  have compl : σ • L'.1.fixingSubgroup.carrier ⊆ H.carrierᶜ :=
+    calc σ • (SetLike.coe L'.1.fixingSubgroup)
+      _ ⊆ σ • (SetLike.coe L.fixingSubgroup) :=
+        Set.smul_set_mono (fixingSubgroup_antitone (L.le_normalClosure))
+      _ ⊆ H.carrierᶜ := Set.smul_set_subset_iff.mpr sub
   have fix : ∀ x ∈ IntermediateField.fixedField H.toSubgroup ⊓ ↑L', σ x = x :=
     fun x hx ↦ ((mem_fixingSubgroup_iff Gal(K/k)).mp hσ) x hx.1
   rw [restrict_fixedField H.1 L'.1] at fix
@@ -195,7 +173,7 @@ lemma fixingSubgroup_fixedField (H : ClosedSubgroup Gal(K/k)) [IsGalois k K] :
 /-- The Galois correspondence from intermediate fields to closed subgroups. -/
 def IntermediateFieldEquivClosedSubgroup [IsGalois k K] :
     IntermediateField k K ≃o (ClosedSubgroup Gal(K/k))ᵒᵈ where
-  toFun L := ⟨L.fixingSubgroup, fixingSubgroup_isClosed L⟩
+  toFun L := ⟨L.fixingSubgroup, L.isClosed_fixingSubgroup⟩
   invFun H := IntermediateField.fixedField H.1
   left_inv L := fixedField_fixingSubgroup L
   right_inv H := by
@@ -208,7 +186,7 @@ def IntermediateFieldEquivClosedSubgroup [IsGalois k K] :
 /-- The Galois correspondence as a `GaloisInsertion` -/
 def GaloisInsertionIntermediateFieldClosedSubgroup [IsGalois k K] :
     GaloisInsertion (OrderDual.toDual ∘ fun (E : IntermediateField k K) ↦
-      (⟨E.fixingSubgroup, fixingSubgroup_isClosed E⟩ : ClosedSubgroup Gal(K/k)))
+      (⟨E.fixingSubgroup, E.isClosed_fixingSubgroup⟩ : ClosedSubgroup Gal(K/k)))
       ((fun (H : ClosedSubgroup Gal(K/k)) ↦ IntermediateField.fixedField H) ∘
         OrderDual.toDual) :=
   OrderIso.toGaloisInsertion IntermediateFieldEquivClosedSubgroup
@@ -240,13 +218,11 @@ set_option backward.isDefEq.respectTransparency false in
 open IntermediateField in
 theorem isOpen_iff_finite (L : IntermediateField k K) [IsGalois k K] :
     IsOpen L.fixingSubgroup.carrier ↔ FiniteDimensional k L := by
-  refine ⟨fun h ↦ ?_, fun h ↦ IntermediateField.fixingSubgroup_isOpen L⟩
+  refine ⟨fun h ↦ ?_, fun h ↦ IntermediateField.isOpen_fixingSubgroup L⟩
   have : (IntermediateFieldEquivClosedSubgroup.toFun L).carrier ∈ nhds 1 :=
     IsOpen.mem_nhds h (congrFun rfl)
-  rw [GroupFilterBasis.nhds_one_eq] at this
-  rcases this with ⟨S, ⟨gp, ⟨M, hM, eq'⟩, eq⟩, sub⟩
-  rw [← eq, ← eq'] at sub
-  have := hM.out
+  rw [krullTopology_mem_nhds_one_iff] at this
+  rcases this with ⟨M, hM, sub⟩
   let L' : FiniteGaloisIntermediateField k K := {
     normalClosure k M K with
     finiteDimensional := normalClosure.is_finiteDimensional k M K

Mathlib/FieldTheory/Galois/Profinite.lean

@@ -163,7 +163,7 @@ theorem restrictNormalHom_continuous (L : IntermediateField k K) [Normal k L] :
     dsimp at this
     rw [hx y.1 ((IntermediateField.mem_lift y).mpr hy)] at this
     exact SetLike.coe_eq_coe.mp this
-  · exact ⟨IntermediateField.fixingSubgroup_isOpen (IntermediateField.lift L'), congrFun rfl⟩
+  · exact ⟨IntermediateField.isOpen_fixingSubgroup (IntermediateField.lift L'), one_mem _⟩
 
 set_option backward.isDefEq.respectTransparency false in
 lemma algEquivToLimit_continuous : Continuous (algEquivToLimit k K) := by
@@ -326,9 +326,6 @@ noncomputable def continuousMulEquivToLimit [IsGalois k K] :
   continuous_toFun := algEquivToLimit_continuous
   continuous_invFun := mulEquivToLimit_symm_continuous
 
-instance [IsGalois k K] : CompactSpace Gal(K/k) :=
-  (continuousMulEquivToLimit k K).symm.compactSpace
-
 /-- `Gal(K/k)` as a profinite group as there is
 a `ContinuousMulEquiv` to a `ProfiniteGrp` given above -/
 noncomputable def profiniteGalGrp [IsGalois k K] : ProfiniteGrp :=

Mathlib/FieldTheory/IntermediateField/Adjoin/Algebra.lean

@@ -93,7 +93,6 @@ lemma fg_top_iff :
     (⊤ : IntermediateField F E).FG ↔ Algebra.EssFiniteType F E := by
   constructor
   · intro ⟨s, hs⟩
-    have : Algebra.FiniteType F (Algebra.adjoin F (s : Set E)) := .adjoin_of_finite s.finite_toSet
     have : Algebra.EssFiniteType (Algebra.adjoin F (s : Set E)) (adjoin F (s : Set E)) :=
       .of_isLocalization _ (nonZeroDivisors _)
     have : Algebra.EssFiniteType F (adjoin F (s : Set E)) :=
@@ -124,6 +123,9 @@ lemma essFiniteType_iff {K : IntermediateField F E} :
       adjoin_map, ← Set.range_comp, Function.comp_def, ← AlgHom.fieldRange_eq_map] using! this
   exact ⟨fun ⟨s, _, hs⟩ ↦ ⟨s, hs⟩, fun ⟨s, hs⟩ ↦ ⟨s, hs ▸ subset_adjoin _ _, hs⟩⟩
 
+instance [Finite S] : Algebra.EssFiniteType F (adjoin F S) :=
+  essFiniteType_iff.mpr (fg_adjoin_of_finite S.toFinite)
+
 /-- A field is finitely generated if and only if it is essentially of finite type over its prime
 subfield. -/
 theorem _root_.Field.fg_iff_essFiniteType : Field.FG F ↔ Algebra.EssFiniteType (⊥ : Subfield F) F :=

Mathlib/FieldTheory/IntermediateField/Adjoin/Basic.lean

@@ -614,6 +614,10 @@ theorem finiteDimensional_adjoin {S : Set L} [Finite S] (hS : ∀ x ∈ S, IsInt
   have (x : S) := adjoin.finiteDimensional (hS x.1 x.2)
   exact finiteDimensional_iSup_of_finite
 
+instance {S : Set L} [Finite S] [Algebra.IsIntegral K L] :
+    FiniteDimensional K (adjoin K S) :=
+  finiteDimensional_adjoin fun _ _ => Algebra.IsIntegral.isIntegral _
+
 /-- If `x` generates `L` over `K` (i.e., `K⟮x⟯ = ⊤`) and is integral over `K`, then `x`
 defines a `PowerBasis` for `L` over `K`. See `PowerBasis.ofAdjoinEqTop` for a version with
 `Algebra.adjoin`. -/