mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.FieldTheory.Galois.Basic

Goal state before:
pp: F : Type u_1
inst✝⁶ : Field F
E : Type u_2
inst✝⁵ : Field E
inst✝⁴ : Algebra F E
p : F[X]
hFE : FiniteDimensional F E
sp : Polynomial.IsSplittingField F E p
hp : p.Separable
K : Type u_3
inst✝³ : Field K
inst✝² : Algebra F K
inst✝¹ : Algebra K E
inst✝ : IsScalarTower F K E
x : E
hx : x ∈ p.aroots E
h : IsIntegral K x
h1 : p ≠ 0
h2 : minpoly K x ∣ Polynomial.map (algebraMap F K) p
key_equiv : (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] E) ≃ (f : K →ₐ[F] E) × (↥K⟮x⟯ →ₐ[K] E) :=
  id algHomEquivSigma
this✝ : ∀ (f : K →ₐ[F] E), Finite (↥K⟮x⟯ →ₐ[K] E)
this : FiniteDimensional F K
f : K →ₐ[F] E
a✝ : f ∈ Finset.univ
⊢ Nat.card (↥K⟮x⟯ →ₐ[K] E) = (minpoly K x).natDegree

Tactic:
← @IntermediateField.card_algHom_adjoin_integral K _ E _ _ x E _ (RingHom.toAlgebra f) h

Goal state after:
F : Type u_1
inst✝⁶ : Field F
E : Type u_2
inst✝⁵ : Field E
inst✝⁴ : Algebra F E
p : F[X]
hFE : FiniteDimensional F E
sp : Polynomial.IsSplittingField F E p
hp : p.Separable
K : Type u_3
inst✝³ : Field K
inst✝² : Algebra F K
inst✝¹ : Algebra K E
inst✝ : IsScalarTower F K E
x : E
hx : x ∈ p.aroots E
h : IsIntegral K x
h1 : p ≠ 0
h2 : minpoly K x ∣ Polynomial.map (algebraMap F K) p
key_equiv : (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] E) ≃ (f : K →ₐ[F] E) × (↥K⟮x⟯ →ₐ[K] E) :=
  id algHomEquivSigma
this✝ : ∀ (f : K →ₐ[F] E), Finite (↥K⟮x⟯ →ₐ[K] E)
this : FiniteDimensional F K
f : K →ₐ[F] E
a✝ : f ∈ Finset.univ
⊢ Nat.card (↥K⟮x⟯ →ₐ[K] E) = Nat.card (↥K⟮x⟯ →ₐ[K] E)

case h_sep
F : Type u_1
inst✝⁶ : Field F
E : Type u_2
inst✝⁵ : Field E
inst✝⁴ : Algebra F E
p : F[X]
hFE : FiniteDimensional F E
sp : Polynomial.IsSplittingField F E p
hp : p.Separable
K : Type u_3
inst✝³ : Field K
inst✝² : Algebra F K
inst✝¹ : Algebra K E
inst✝ : IsScalarTower F K E
x : E
hx : x ∈ p.aroots E
h : IsIntegral K x
h1 : p ≠ 0
h2 : minpoly K x ∣ Polynomial.map (algebraMap F K) p
key_equiv : (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] E) ≃ (f : K →ₐ[F] E) × (↥K⟮x⟯ →ₐ[K] E) :=
  id algHomEquivSigma
this✝ : ∀ (f : K →ₐ[F] E), Finite (↥K⟮x⟯ →ₐ[K] E)
this : FiniteDimensional F K
f : K →ₐ[F] E
a✝ : f ∈ Finset.univ
⊢ IsSeparable K x

case h_splits
F : Type u_1
inst✝⁶ : Field F
E : Type u_2
inst✝⁵ : Field E
inst✝⁴ : Algebra F E
p : F[X]
hFE : FiniteDimensional F E
sp : Polynomial.IsSplittingField F E p
hp : p.Separable
K : Type u_3
inst✝³ : Field K
inst✝² : Algebra F K
inst✝¹ : Algebra K E
inst✝ : IsScalarTower F K E
x : E
hx : x ∈ p.aroots E
h : IsIntegral K x
h1 : p ≠ 0
h2 : minpoly K x ∣ Polynomial.map (algebraMap F K) p
key_equiv : (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] E) ≃ (f : K →ₐ[F] E) × (↥K⟮x⟯ →ₐ[K] E) :=
  id algHomEquivSigma
this✝ : ∀ (f : K →ₐ[F] E), Finite (↥K⟮x⟯ →ₐ[K] E)
this : FiniteDimensional F K
f : K →ₐ[F] E
a✝ : f ∈ Finset.univ
⊢ (Polynomial.map (algebraMap K E) (minpoly K x)).Splits
Module: Mathlib.FieldTheory.Galois.Basic

Goal state before:
pp: case h_splits
F : Type u_1
inst✝⁶ : Field F
E : Type u_2
inst✝⁵ : Field E
inst✝⁴ : Algebra F E
p : F[X]
hFE : FiniteDimensional F E
sp : Polynomial.IsSplittingField F E p
hp : p.Separable
K : Type u_3
inst✝³ : Field K
inst✝² : Algebra F K
inst✝¹ : Algebra K E
inst✝ : IsScalarTower F K E
x : E
hx : x ∈ p.aroots E
h : IsIntegral K x
h1 : p ≠ 0
h2 : minpoly K x ∣ Polynomial.map (algebraMap F K) p
key_equiv : (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] E) ≃ (f : K →ₐ[F] E) × (↥K⟮x⟯ →ₐ[K] E) :=
  id algHomEquivSigma
this✝ : ∀ (f : K →ₐ[F] E), Finite (↥K⟮x⟯ →ₐ[K] E)
this : FiniteDimensional F K
f : K →ₐ[F] E
a✝ : f ∈ Finset.univ
⊢ (Polynomial.map (algebraMap K E) (minpoly K x)).Splits

Tactic:
· apply sp.splits.of_dvd (Polynomial.map_ne_zero h1)
    rwa [← f.comp_algebraMap, ← p.map_map, RingHom.algebraMap_toAlgebra, Polynomial.map_dvd_map']

Goal state after:
Module: Mathlib.FieldTheory.SeparableDegree

Goal state before:
pp: F : Type u
E : Type v
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
x : E
⊢ Algebra.IsSeparable F ↥F⟮x⟯ ↔ IsSeparable F x

Tactic:
refine ⟨fun _ ↦ ?_, fun hsep ↦ ?_⟩

Goal state after:
case refine_1
F : Type u
E : Type v
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
x : E
x✝ : Algebra.IsSeparable F ↥F⟮x⟯
⊢ IsSeparable F x

case refine_2
F : Type u
E : Type v
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
x : E
hsep : IsSeparable F x
⊢ Algebra.IsSeparable F ↥F⟮x⟯
Module: Mathlib.Topology.Algebra.OpenSubgroup

Goal state before:
pp: G : Type u_2
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalGroup G
inst✝ : CompactSpace G
W : Set G
WClopen : IsClopen W
einW : 1 ∈ W
V : Set G
hV : mulInvClosureNhd V W
S : Subgroup G := { carrier := ⋃ n, V ^ (n + 1), mul_mem' := ⋯, one_mem' := ⋯, inv_mem' := ⋯ }
this : IsOpen[inst✝²] (⋃ n, V ^ (n + 1))
mulVpow : ∀ (n : ℕ), W * V ^ (n + 1) ⊆ W
n : ℕ
x : G
⊢ x ∈ V ^ (n + 1) → x ∈ W * V ^ (n + 1)

Tactic:
xin

Goal state after:
G : Type u_2
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalGroup G
inst✝ : CompactSpace G
W : Set G
WClopen : IsClopen W
einW : 1 ∈ W
V : Set G
hV : mulInvClosureNhd V W
S : Subgroup G := { carrier := ⋃ n, V ^ (n + 1), mul_mem' := ⋯, one_mem' := ⋯, inv_mem' := ⋯ }
this : IsOpen[inst✝²] (⋃ n, V ^ (n + 1))
mulVpow : ∀ (n : ℕ), W * V ^ (n + 1) ⊆ W
n : ℕ
x : G
xin : x ∈ V ^ (n + 1)
⊢ x ∈ W * V ^ (n + 1)
Module: Mathlib.FieldTheory.KrullTopology

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : Algebra.IsIntegral K L
f g : Gal(L/K)
hfg : f ≠ g
φ : Gal(L/K) := f⁻¹ * g
x : L
hx : f x ≠ g x
hφx : φ x ≠ x
E : IntermediateField K L := K⟮x⟯
⊢ (fun x y ↦ ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v) f g

Tactic:
let h_findim : FiniteDimensional K E := IntermediateField.adjoin.finiteDimensional
        (Algebra.IsIntegral.isIntegral x)

Goal state after:
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : Algebra.IsIntegral K L
f g : Gal(L/K)
hfg : f ≠ g
φ : Gal(L/K) := f⁻¹ * g
x : L
hx : f x ≠ g x
hφx : φ x ≠ x
E : IntermediateField K L := K⟮x⟯
h_findim : FiniteDimensional K ↥E := IntermediateField.adjoin.finiteDimensional (Algebra.IsIntegral.isIntegral x)
⊢ (fun x y ↦ ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v) f g
Module: Mathlib.FieldTheory.KrullTopology

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : Algebra.IsIntegral K L
f g : Gal(L/K)
hfg : f ≠ g
φ : Gal(L/K) := f⁻¹ * g
x : L
hx : f x ≠ g x
hφx : φ x ≠ x
E : IntermediateField K L := K⟮x⟯
h_findim : FiniteDimensional K ↥E := IntermediateField.adjoin.finiteDimensional (Algebra.IsIntegral.isIntegral x)
H : Subgroup Gal(L/K) := E.fixingSubgroup
h_basis : ↑H ∈ galGroupBasis K L
W : Set Gal(L/K)
hWH : W ⊆ ↑H
hW_open : IsOpen W
hW_1 : 1 ∈ W
⊢ ∀ ⦃a : Gal(L/K)⦄, a ∈ f • W → a ∉ g • W

Tactic:
rintro σ ⟨w1, hw1, h⟩ ⟨w2, hw2, rfl⟩

Goal state after:
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : Algebra.IsIntegral K L
f g : Gal(L/K)
hfg : f ≠ g
φ : Gal(L/K) := f⁻¹ * g
x : L
hx : f x ≠ g x
hφx : φ x ≠ x
E : IntermediateField K L := K⟮x⟯
h_findim : FiniteDimensional K ↥E := IntermediateField.adjoin.finiteDimensional (Algebra.IsIntegral.isIntegral x)
H : Subgroup Gal(L/K) := E.fixingSubgroup
h_basis : ↑H ∈ galGroupBasis K L
W : Set Gal(L/K)
hWH : W ⊆ ↑H
hW_open : IsOpen W
hW_1 : 1 ∈ W
w1 : Gal(L/K)
hw1 : w1 ∈ W
w2 : Gal(L/K)
hw2 : w2 ∈ W
h : (fun x ↦ f • x) w1 = (fun x ↦ g • x) w2
⊢ False
Module: Mathlib.FieldTheory.KrullTopology

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : Algebra.IsIntegral K L
σ τ : Gal(L/K)
h_diff : σ ≠ τ
hστ : σ⁻¹ * τ ≠ 1
⊢ ∃ U, IsClopen U ∧ σ ∈ U ∧ τ ∈ Uᶜ

Tactic:
rcases DFunLike.exists_ne hστ with ⟨x, hx : (σ⁻¹ * τ) x ≠ x⟩

Goal state after:
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : Algebra.IsIntegral K L
σ τ : Gal(L/K)
h_diff : σ ≠ τ
hστ : σ⁻¹ * τ ≠ 1
x : L
hx : (σ⁻¹ * τ) x ≠ x
⊢ ∃ U, IsClopen U ∧ σ ∈ U ∧ τ ∈ Uᶜ
Module: Mathlib.LinearAlgebra.BilinearForm.DualLattice

Goal state before:
pp: case a
R : Type u_4
S : Type u_2
M : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : Field S
inst✝⁶ : AddCommGroup M
inst✝⁵ : Algebra R S
inst✝⁴ : Module R M
inst✝³ : Module S M
inst✝² : IsScalarTower R S M
B : BilinForm S M
ι : Type u_1
inst✝¹ : Finite ι
inst✝ : DecidableEq ι
hB : B.Nondegenerate
b : Basis ι S M
val✝ : Fintype ι
i : ι
f : ι → R
hy : ∑ i, f i • b i ∈ Submodule.span R (Set.range ⇑b)
j : ι
⊢ ((algebraMap R S) (f j) • if j = i then 1 else 0) ∈ 1

Tactic:
smul_eq_mul,

Goal state after:
case a
R : Type u_4
S : Type u_2
M : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : Field S
inst✝⁶ : AddCommGroup M
inst✝⁵ : Algebra R S
inst✝⁴ : Module R M
inst✝³ : Module S M
inst✝² : IsScalarTower R S M
B : BilinForm S M
ι : Type u_1
inst✝¹ : Finite ι
inst✝ : DecidableEq ι
hB : B.Nondegenerate
b : Basis ι S M
val✝ : Fintype ι
i : ι
f : ι → R
hy : ∑ i, f i • b i ∈ Submodule.span R (Set.range ⇑b)
j : ι
⊢ ((algebraMap R S) (f j) * if j = i then 1 else 0) ∈ 1
Module: Mathlib.FieldTheory.Galois.Infinite

Goal state before:
pp: k : Type u_1
K : Type u_2
inst✝³ : Field k
inst✝² : Field K
inst✝¹ : Algebra k K
L : IntermediateField k K
inst✝ : IsGalois k K
x : K
hx : ∀ f ∈ L.fixingSubgroup, f x = x
mem : x ∈ (adjoin ↥L {x}).toIntermediateField
this : IntermediateField.fixedField ⊤ = ⊥
⊢ ⟨x, mem⟩ ∈ ⊥

Tactic:
← this,

Goal state after:
k : Type u_1
K : Type u_2
inst✝³ : Field k
inst✝² : Field K
inst✝¹ : Algebra k K
L : IntermediateField k K
inst✝ : IsGalois k K
x : K
hx : ∀ f ∈ L.fixingSubgroup, f x = x
mem : x ∈ (adjoin ↥L {x}).toIntermediateField
this : IntermediateField.fixedField ⊤ = ⊥
⊢ ⟨x, mem⟩ ∈ IntermediateField.fixedField ⊤
Module: Mathlib.FieldTheory.Galois.Infinite

Goal state before:
pp: k : Type u_1
K : Type u_2
inst✝³ : Field k
inst✝² : Field K
inst✝¹ : Algebra k K
L : IntermediateField k K
inst✝ : IsGalois k K
x : K
hx : ∀ f ∈ L.fixingSubgroup, f x = x
mem : x ∈ (adjoin ↥L {x}).toIntermediateField
this : IntermediateField.fixedField ⊤ = ⊥
f : Gal(↥(adjoin ↥L {x}).toIntermediateField/↥L)
a✝ : f ∈ ⊤
σ : Gal(K/↥L)
hσ : (restrictNormalHom ↥(adjoin ↥L {x}).toIntermediateField) σ = f
⊢ f ⟨x, mem⟩ = ⟨x, mem⟩

Tactic:
apply Subtype.val_injective

Goal state after:
k : Type u_1
K : Type u_2
inst✝³ : Field k
inst✝² : Field K
inst✝¹ : Algebra k K
L : IntermediateField k K
inst✝ : IsGalois k K
x : K
hx : ∀ f ∈ L.fixingSubgroup, f x = x
mem : x ∈ (adjoin ↥L {x}).toIntermediateField
this : IntermediateField.fixedField ⊤ = ⊥
f : Gal(↥(adjoin ↥L {x}).toIntermediateField/↥L)
a✝ : f ∈ ⊤
σ : Gal(K/↥L)
hσ : (restrictNormalHom ↥(adjoin ↥L {x}).toIntermediateField) σ = f
⊢ ↑(f ⟨x, mem⟩) = ↑⟨x, mem⟩
Module: Mathlib.FieldTheory.Galois.Infinite

Goal state before:
pp: case refine_1
k : Type u_1
K : Type u_2
inst✝³ : Field k
inst✝² : Field K
inst✝¹ : Algebra k K
H : Subgroup Gal(K/k)
L : IntermediateField k K
inst✝ : Normal k ↥L
x : K
h : x ∈ ↑(fixedField H ⊓ L)
xL : x ∈ (fun x ↦ ↑x) L
σ : Gal(K/k)
hσ : σ ∈ H
⊢ ((restrictNormalHom ↥L) σ) ⟨x, xL⟩ = ⟨x, xL⟩

Tactic:
apply Subtype.val_injective

Goal state after:
case refine_1
k : Type u_1
K : Type u_2
inst✝³ : Field k
inst✝² : Field K
inst✝¹ : Algebra k K
H : Subgroup Gal(K/k)
L : IntermediateField k K
inst✝ : Normal k ↥L
x : K
h : x ∈ ↑(fixedField H ⊓ L)
xL : x ∈ (fun x ↦ ↑x) L
σ : Gal(K/k)
hσ : σ ∈ H
⊢ ↑(((restrictNormalHom ↥L) σ) ⟨x, xL⟩) = ↑⟨x, xL⟩
Module: Mathlib.FieldTheory.Minpoly.MinpolyDiv

Goal state before:
pp: R : Type u_2
S : Type u_1
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
hx : IsIntegral R x
a✝ : Nontrivial S
this : (minpolyDiv R x).leadingCoeff * (X - C x).leadingCoeff = 1
⊢ (minpolyDiv R x).Monic

Tactic:
simpa using! this

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain R
inst✝³ : Algebra R S
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDomain S
inst✝ : IsTorsionFree R S
x : S
hx : IsIntegral R x
a b : R[X]
h : minpoly R x ∣ a * b
h' : ¬minpoly R x ∣ a
⊢ minpoly R x ∣ b

Tactic:
rw [← minpoly.isIntegrallyClosed_dvd_iff hx] at h' h ⊢

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain R
inst✝³ : Algebra R S
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDomain S
inst✝ : IsTorsionFree R S
x : S
hx : IsIntegral R x
a b : R[X]
h : (Polynomial.aeval x) (a * b) = 0
h' : ¬(Polynomial.aeval x) a = 0
⊢ (Polynomial.aeval x) b = 0
Module: Mathlib.FieldTheory.Galois.Infinite

Goal state before:
pp: k : Type u_1
K : Type u_2
inst✝³ : Field k
inst✝² : Field K
inst✝¹ : Algebra k K
H : ClosedSubgroup Gal(K/k)
inst✝ : IsGalois k K
σ : Gal(K/k)
hσ : σ ∈ (fixedField ↑H).fixingSubgroup
h : σ ∉ ↑H
b : Set Gal(K/k)
sub : b ⊆ (fun y ↦ σ * y) ⁻¹' (↑H).carrierᶜ
gp : Subgroup Gal(K/k)
L : IntermediateField k K
eq : (fun g ↦ g.carrier) L.fixingSubgroup = b
hL : L ∈ finiteExts k K
eq' : L.fixingSubgroup = gp
this : FiniteDimensional k ↥L
L' : FiniteGaloisIntermediateField k K :=
  let __src := normalClosure k (↥L) K;
  { toIntermediateField := __src, finiteDimensional := ⋯, isGalois := ⋯ }
compl : σ • L'.fixingSubgroup.carrier ⊆ (↑H).carrierᶜ
fix : ∀ x ∈ lift (fixedField (Subgroup.map (restrictNormalHom ↥L'.toIntermediateField) ↑H)), σ x = x
y : ↥L'.toIntermediateField
hy : y ∈ ↑(fixedField (Subgroup.map (restrictNormalHom ↥L'.toIntermediateField) ↑H))
⊢ (restrictNormalHom ↥L'.toIntermediateField) σ • y = y

Tactic:
apply Subtype.val_injective

Goal state after:
k : Type u_1
K : Type u_2
inst✝³ : Field k
inst✝² : Field K
inst✝¹ : Algebra k K
H : ClosedSubgroup Gal(K/k)
inst✝ : IsGalois k K
σ : Gal(K/k)
hσ : σ ∈ (fixedField ↑H).fixingSubgroup
h : σ ∉ ↑H
b : Set Gal(K/k)
sub : b ⊆ (fun y ↦ σ * y) ⁻¹' (↑H).carrierᶜ
gp : Subgroup Gal(K/k)
L : IntermediateField k K
eq : (fun g ↦ g.carrier) L.fixingSubgroup = b
hL : L ∈ finiteExts k K
eq' : L.fixingSubgroup = gp
this : FiniteDimensional k ↥L
L' : FiniteGaloisIntermediateField k K := ⋯
compl : σ • L'.fixingSubgroup.carrier ⊆ (↑H).carrierᶜ
fix : ∀ x ∈ lift (fixedField (Subgroup.map (restrictNormalHom ↥L'.toIntermediateField) ↑H)), σ x = x
y : ↥L'.toIntermediateField
hy : y ∈ ↑(fixedField (Subgroup.map (restrictNormalHom ↥L'.toIntermediateField) ↑H))
⊢ ↑((restrictNormalHom ↥L'.toIntermediateField) σ • y) = ↑y
Module: Mathlib.RingTheory.Polynomial.GaussLemma

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : IsGCDMonoid R
f : R[X]
hf : f.IsPrimitive
g : K[X]
⊢ g * map (algebraMap R K) f ∈ lifts (algebraMap R K) ↔ g ∈ lifts (algebraMap R K)

Tactic:
let : NormalizedGCDMonoid R := Nonempty.some inferInstance

Goal state after:
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : IsGCDMonoid R
f : R[X]
hf : f.IsPrimitive
g : K[X]
this : NormalizedGCDMonoid R := ⋯.some
⊢ g * map (algebraMap R K) f ∈ lifts (algebraMap R K) ↔ g ∈ lifts (algebraMap R K)
Module: Mathlib.FieldTheory.IsSepClosed

Goal state before:
pp: F : Type u_1
E : Type u_2
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : IsSepClosed E
⊢ separableClosure F E = ⊥ ↔ IsSepClosed F

Tactic:
refine ⟨fun h ↦ IsSepClosed.of_exists_root _ fun p _ hirr hsep ↦ ?_,
    fun _ ↦ IntermediateField.eq_bot_of_isSepClosed_of_isSeparable _⟩

Goal state after:
F : Type u_1
E : Type u_2
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : IsSepClosed E
h : separableClosure F E = ⊥
p : F[X]
x✝ : p.Monic
hirr : Irreducible p
hsep : p.Separable
⊢ ∃ x, eval x p = 0
Module: Mathlib.RingTheory.Polynomial.GaussLemma

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : IsGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
⊢ IsUnit a ∨ IsUnit b

Tactic:
have h1 : C c * C d * p = integerNormalization R⁰ a * integerNormalization R⁰ b := by
    apply map_injective (algebraMap R K) (IsFractionRing.injective _ _) _
    rw [Polynomial.map_mul, Polynomial.map_mul, Polynomial.map_mul, hc, hd, map_C, map_C, hab]
    ring

Goal state after:
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : IsGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
h1 : C c * C d * p = integerNormalization R⁰ a * integerNormalization R⁰ b
⊢ IsUnit a ∨ IsUnit b
Module: Mathlib.FieldTheory.PurelyInseparable.Basic

Goal state before:
pp: F✝ : Type u
E✝ : Type v
inst✝¹¹ : Field F✝
inst✝¹⁰ : Field E✝
inst✝⁹ : Algebra F✝ E✝
q✝ : ℕ
inst✝⁸ : ExpChar F✝ q✝
inst✝⁷ : IsPurelyInseparable F✝ E✝
inst✝⁶ : FiniteDimensional F✝ E✝
E : Type v
inst✝⁵ : Field E
q d : ℕ
IH :
  ∀ m < d,
    ∀ (F : Type v) [inst : Field F] [inst_1 : Algebra F E] [ExpChar F q] [IsPurelyInseparable F E]
      [FiniteDimensional F E], finrank F E = m → ∃ n, m = q ^ n
F : Type v
inst✝⁴ : Field F
inst✝³ : Algebra F E
inst✝² : ExpChar F q
inst✝¹ : IsPurelyInseparable F E
inst✝ : FiniteDimensional F E
hd : finrank F E = d
h : ¬⊥ = ⊤
x : E
hx : x ∉ ⊥
m : ℕ
y : F
e : minpoly F x = X ^ q ^ m - C y
this : finrank F ↥F⟮x⟯ = q ^ m
⊢ 1 < q ^ m

Tactic:
by_contra! H

Goal state after:
F✝ : Type u
E✝ : Type v
inst✝¹¹ : Field F✝
inst✝¹⁰ : Field E✝
inst✝⁹ : Algebra F✝ E✝
q✝ : ℕ
inst✝⁸ : ExpChar F✝ q✝
inst✝⁷ : IsPurelyInseparable F✝ E✝
inst✝⁶ : FiniteDimensional F✝ E✝
E : Type v
inst✝⁵ : Field E
q d : ℕ
IH :
  ∀ m < d,
    ∀ (F : Type v) [inst : Field F] [inst_1 : Algebra F E] [ExpChar F q] [IsPurelyInseparable F E]
      [FiniteDimensional F E], finrank F E = m → ∃ n, m = q ^ n
F : Type v
inst✝⁴ : Field F
inst✝³ : Algebra F E
inst✝² : ExpChar F q
inst✝¹ : IsPurelyInseparable F E
inst✝ : FiniteDimensional F E
hd : finrank F E = d
h : ¬⊥ = ⊤
x : E
hx : x ∉ ⊥
m : ℕ
y : F
e : minpoly F x = X ^ q ^ m - C y
this : finrank F ↥F⟮x⟯ = q ^ m
H : q ^ m ≤ 1
⊢ False
Module: Mathlib.RingTheory.DedekindDomain.IntegralClosure

Goal state before:
pp: case refine_2
A : Type u_1
K : Type u_2
inst✝¹⁴ : CommRing A
inst✝¹³ : Field K
inst✝¹² : Algebra A K
inst✝¹¹ : IsFractionRing A K
L : Type u_3
inst✝¹⁰ : Field L
C : Type u_4
inst✝⁹ : CommRing C
inst✝⁸ : Algebra K L
inst✝⁷ : Algebra A L
inst✝⁶ : IsScalarTower A K L
inst✝⁵ : Algebra C L
inst✝⁴ : IsIntegralClosure C A L
inst✝³ : Algebra A C
inst✝² : IsScalarTower A C L
inst✝¹ : IsDomain A
inst✝ : Algebra.IsAlgebraic K L
this✝¹ : IsDomain C
this✝ : IsTorsionFree A L
this : IsTorsionFree A C
z : L
m : ↥A⁰
hm : IsIntegral A (m • z)
⊢ ∃ x, z * (algebraMap C L) ↑x.2 = (algebraMap C L) x.1

Tactic:
obtain ⟨x, hx⟩ : ∃ x, algebraMap C L x = m • z := IsIntegralClosure.isIntegral_iff.mp hm

Goal state after:
case refine_2
A : Type u_1
K : Type u_2
inst✝¹⁴ : CommRing A
inst✝¹³ : Field K
inst✝¹² : Algebra A K
inst✝¹¹ : IsFractionRing A K
L : Type u_3
inst✝¹⁰ : Field L
C : Type u_4
inst✝⁹ : CommRing C
inst✝⁸ : Algebra K L
inst✝⁷ : Algebra A L
inst✝⁶ : IsScalarTower A K L
inst✝⁵ : Algebra C L
inst✝⁴ : IsIntegralClosure C A L
inst✝³ : Algebra A C
inst✝² : IsScalarTower A C L
inst✝¹ : IsDomain A
inst✝ : Algebra.IsAlgebraic K L
this✝¹ : IsDomain C
this✝ : IsTorsionFree A L
this : IsTorsionFree A C
z : L
m : ↥A⁰
hm : IsIntegral A (m • z)
x : C
hx : (algebraMap C L) x = m • z
⊢ ∃ x, z * (algebraMap C L) ↑x.2 = (algebraMap C L) x.1
Module: Mathlib.FieldTheory.PurelyInseparable.Basic

Goal state before:
pp: k : Type u_1
K : Type u_2
R : Type u_3
inst✝⁶ : Field k
inst✝⁵ : Field K
inst✝⁴ : Algebra k K
inst✝³ : CommRing R
inst✝² : Algebra k R
inst✝¹ : IsPurelyInseparable k K
q : ℕ
inst✝ : ExpChar k q
x : R ⊗[k] K
⊢ ∃ n, x ^ q ^ n ∈ (algebraMap R (R ⊗[k] K)).range

Tactic:
by
  nontriviality (R ⊗[k] K)
  obtain (hq | hq) := expChar_is_prime_or_one k q
  induction x with
  | zero => exact ⟨0, 0, by simp⟩
  | add x y h h' =>
    have : ExpChar (R ⊗[k] K) q := expChar_of_injective_ringHom (algebraMap k _).injective q
    simp_rw [RingHom.mem_range, ← RingHom.mem_rangeS, ← Subalgebra.mem_perfectClosure_iff] at h h' ⊢
    exact add_mem h h'
  | tmul x y =>
    obtain ⟨n, a, ha⟩ := IsPurelyInseparable.pow_mem k q y
    use n
    have : (x ^ q ^ n) ⊗ₜ[k] (y ^ q ^ n) =
        (x ^ q ^ n) ⊗ₜ[k] (1 : K) * (1 : R) ⊗ₜ[k] (y ^ q ^ n) := by
      rw [Algebra.TensorProduct.tmul_mul_tmul, mul_one, one_mul]
    rw [Algebra.TensorProduct.tmul_pow, this]
    refine Subring.mul_mem _ ⟨x ^ q ^ n, rfl⟩ ⟨algebraMap k R a, ?_⟩
    rw [← IsScalarTower.algebraMap_apply, Algebra.TensorProduct.algebraMap_apply,
      Algebra.TensorProduct.tmul_one_eq_one_tmul, ha]
  · subst hq
    have : CharZero k := charZero_of_expChar_one' k
    exact ⟨0, (Algebra.TensorProduct.includeLeft_surjective R _ <|
      IsPurelyInseparable.surjective_algebraMap_of_isSeparable k K) _⟩

Goal state after:
Module: Mathlib.RingTheory.Trace.Basic

Goal state before:
pp: K : Type u_4
L : Type u_5
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
x : L
hx : ¬IsIntegral K x
⊢ (Algebra.trace K ↥K⟮x⟯) (gen K x) = 0

Tactic:
rw [trace_eq_zero_of_not_exists_basis, LinearMap.zero_apply]

Goal state after:
case h
K : Type u_4
L : Type u_5
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
x : L
hx : ¬IsIntegral K x
⊢ ¬∃ s, Nonempty (Basis (↥s) K ↥K⟮x⟯)
Module: Mathlib.RingTheory.IsGaloisGroup.Basic

Goal state before:
pp: G : Type u_1
inst✝⁹ : Group G
H : Subgroup G
A : Type u_2
B : Type u_3
C : Type u_4
inst✝⁸ : CommSemiring A
inst✝⁷ : Semiring C
inst✝⁶ : Algebra A C
inst✝⁵ : MulSemiringAction G C
N : Subgroup G
inst✝⁴ : CommSemiring B
inst✝³ : Algebra B C
inst✝² : FaithfulSMul B C
inst✝¹ : IsGaloisGroup (↥N) B C
inst✝ : N.Normal
this : MulSemiringAction G B := mulSemiringActionOfNormal G B C N
q : G ⧸ N
x : B
g₁ g₂ : G
h : (QuotientGroup.leftRel N) g₁ g₂
⊢ (algebraMap B C) x = (g₁⁻¹ • g₂) • (algebraMap B C) x

Tactic:
smul_eq_mul,

Goal state after:
G : Type u_1
inst✝⁹ : Group G
H : Subgroup G
A : Type u_2
B : Type u_3
C : Type u_4
inst✝⁸ : CommSemiring A
inst✝⁷ : Semiring C
inst✝⁶ : Algebra A C
inst✝⁵ : MulSemiringAction G C
N : Subgroup G
inst✝⁴ : CommSemiring B
inst✝³ : Algebra B C
inst✝² : FaithfulSMul B C
inst✝¹ : IsGaloisGroup (↥N) B C
inst✝ : N.Normal
this : MulSemiringAction G B := mulSemiringActionOfNormal G B C N
q : G ⧸ N
x : B
g₁ g₂ : G
h : (QuotientGroup.leftRel N) g₁ g₂
⊢ (algebraMap B C) x = (g₁⁻¹ * g₂) • (algebraMap B C) x
Module: Mathlib.RingTheory.Invariant.Basic

Goal state before:
pp: A : Type u_1
B : Type u_2
G : Type u_3
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : Group G
inst✝³ : MulSemiringAction G B
inst✝² : IsInvariant A B G
inst✝¹ : Finite G
inst✝ : SMulCommClass G A B
P✝ Q✝ : Ideal B
hP✝ : P✝.IsPrime
hQ✝ : Q✝.IsPrime
hPQ : Ideal.under A P✝ = Ideal.under A Q✝
val✝ : Fintype G
P Q : Ideal B
hP : P.IsPrime
hQ : Q.IsPrime
⊢ Ideal.under A P = Ideal.under A Q → ∃ g ∈ ⊤, Q ≤ g • P

Tactic:
hPQ

Goal state after:
A : Type u_1
B : Type u_2
G : Type u_3
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : Group G
inst✝³ : MulSemiringAction G B
inst✝² : IsInvariant A B G
inst✝¹ : Finite G
inst✝ : SMulCommClass G A B
P✝ Q✝ : Ideal B
hP✝ : P✝.IsPrime
hQ✝ : Q✝.IsPrime
hPQ✝ : Ideal.under A P✝ = Ideal.under A Q✝
val✝ : Fintype G
P Q : Ideal B
hP : P.IsPrime
hQ : Q.IsPrime
hPQ : Ideal.under A P = Ideal.under A Q
⊢ ∃ g ∈ ⊤, Q ≤ g • P