Theorem: IsGalois.of_separable_splitting_field_aux
File: Mathlib/FieldTheory/Galois.lean

Proof steps:
state_before: F : Type u_1
inst✝⁸ : Field F
E : Type u_2
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
p : F[X]
hFE : FiniteDimensional F E
sp : Polynomial.IsSplittingField F E p
hp : p.Separable
K : Type u_3
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra K E
inst✝² : IsScalarTower F K E
x : E
hx : x ∈ p.aroots E
inst✝¹ : Fintype (K →ₐ[F] E)
inst✝ : Fintype (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] E)
⊢ Fintype.card (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] E) = Fintype.card (K →ₐ[F] E) * finrank K ↥K⟮x⟯
state_after: F : Type u_1
inst✝⁸ : Field F
E : Type u_2
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
p : F[X]
hFE : FiniteDimensional F E
sp : Polynomial.IsSplittingField F E p
hp : p.Separable
K : Type u_3
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra K E
inst✝² : IsScalarTower F K E
x : E
hx : x ∈ p.aroots E
inst✝¹ : Fintype (K →ₐ[F] E)
inst✝ : Fintype (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] E)
h : IsIntegral K x
⊢ Fintype.card (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] E) = Fintype.card (K →ₐ[F] E) * finrank K ↥K⟮x⟯
tactic: have h : IsIntegral K x := (isIntegral_of_noetherian (IsNoetherian.iff_fg.2 hFE) x).tower_top
state_before: F : Type u_1
inst✝⁸ : Field F
E : Type u_2
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
p : F[X]
hFE : FiniteDimensional F E
sp : Polynomial.IsSplittingField F E p
hp : p.Separable
K : Type u_3
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra K E
inst✝² : IsScalarTower F K E
x : E
hx : x ∈ p.aroots E
inst✝¹ : Fintype (K →ₐ[F] E)
inst✝ : Fintype (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] E)
h : IsIntegral K x
⊢ Fintype.card (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] E) = Fintype.card (K →ₐ[F] E) * finrank K ↥K⟮x⟯
state_after: F : Type u_1
inst✝⁸ : Field F
E : Type u_2
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
p : F[X]
hFE : FiniteDimensional F E
sp : Polynomial.IsSplittingField F E p
hp : p.Separable
K : Type u_3
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra K E
inst✝² : IsScalarTower F K E
x : E
hx : x ∈ p.aroots E
inst✝¹ : Fintype (K →ₐ[F] E)
inst✝ : Fintype (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] E)
h : IsIntegral K x
h1 : p ≠ 0
⊢ Fintype.card (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] E) = Fintype.card (K →ₐ[F] E) * finrank K ↥K⟮x⟯
tactic: have h1 : p ≠ 0 := fun hp => by
    rw [hp, Polynomial.aroots_zero] at hx
    exact Multiset.not_mem_zero x hx
state_before: F : Type u_1
inst✝⁸ : Field F
E : Type u_2
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
p : F[X]
hFE : FiniteDimensional F E
sp : Polynomial.IsSplittingField F E p
hp : p.Separable
K : Type u_3
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra K E
inst✝² : IsScalarTower F K E
x : E
hx : x ∈ p.aroots E
inst✝¹ : Fintype (K →ₐ[F] E)
inst✝ : Fintype (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] E)
h : IsIntegral K x
h1 : p ≠ 0
⊢ Fintype.card (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] E) = Fintype.card (K →ₐ[F] E) * finrank K ↥K⟮x⟯
state_after: F : Type u_1
inst✝⁸ : Field F
E : Type u_2
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
p : F[X]
hFE : FiniteDimensional F E
sp : Polynomial.IsSplittingField F E p
hp : p.Separable
K : Type u_3
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra K E
inst✝² : IsScalarTower F K E
x : E
hx : x ∈ p.aroots E
inst✝¹ : Fintype (K →ₐ[F] E)
inst✝ : Fintype (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] E)
h : IsIntegral K x
h1 : p ≠ 0
h2 : minpoly K x ∣ Polynomial.map (algebraMap F K) p
⊢ Fintype.card (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] E) = Fintype.card (K →ₐ[F] E) * finrank K ↥K⟮x⟯
tactic: have h2 : minpoly K x ∣ p.map (algebraMap F K) := by
    apply minpoly.dvd
    rw [Polynomial.aeval_def, Polynomial.eval₂_map, ← Polynomial.eval_map, ←
      IsScalarTower.algebraMap_eq]
    exact (Polynomial.mem_roots (Polynomial.map_ne_zero h1)).mp hx
state_before: F : Type u_1
inst✝⁸ : Field F
E : Type u_2
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
p : F[X]
hFE : FiniteDimensional F E
sp : Polynomial.IsSplittingField F E p
hp : p.Separable
K : Type u_3
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra K E
inst✝² : IsScalarTower F K E
x : E
hx : x ∈ p.aroots E
inst✝¹ : Fintype (K →ₐ[F] E)
inst✝ : Fintype (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] E)
h : IsIntegral K x
h1 : p ≠ 0
h2 : minpoly K x ∣ Polynomial.map (algebraMap F K) p
⊢ Fintype.card (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] E) = Fintype.card (K →ₐ[F] E) * finrank K ↥K⟮x⟯
state_after: F : Type u_1
inst✝⁸ : Field F
E : Type u_2
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
p : F[X]
hFE : FiniteDimensional F E
sp : Polynomial.IsSplittingField F E p
hp : p.Separable
K : Type u_3
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra K E
inst✝² : IsScalarTower F K E
x : E
hx : x ∈ p.aroots E
inst✝¹ : Fintype (K →ₐ[F] E)
inst✝ : Fintype (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] 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) :=
  let_fun this := algHomEquivSigma;
  this
⊢ Fintype.card (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] E) = Fintype.card (K →ₐ[F] E) * finrank K ↥K⟮x⟯
tactic: let key_equiv : (K⟮x⟯.restrictScalars F →ₐ[F] E) ≃
      Σ f : K →ₐ[F] E, @AlgHom K K⟮x⟯ E _ _ _ _ (RingHom.toAlgebra f) := by
    change (K⟮x⟯ →ₐ[F] E) ≃ Σ f : K →ₐ[F] E, _
    exact algHomEquivSigma
state_before: F : Type u_1
inst✝⁸ : Field F
E : Type u_2
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
p : F[X]
hFE : FiniteDimensional F E
sp : Polynomial.IsSplittingField F E p
hp : p.Separable
K : Type u_3
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra K E
inst✝² : IsScalarTower F K E
x : E
hx : x ∈ p.aroots E
inst✝¹ : Fintype (K →ₐ[F] E)
inst✝ : Fintype (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] 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) :=
  let_fun this := algHomEquivSigma;
  this
⊢ Fintype.card (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] E) = Fintype.card (K →ₐ[F] E) * finrank K ↥K⟮x⟯
state_after: F : Type u_1
inst✝⁸ : Field F
E : Type u_2
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
p : F[X]
hFE : FiniteDimensional F E
sp : Polynomial.IsSplittingField F E p
hp : p.Separable
K : Type u_3
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra K E
inst✝² : IsScalarTower F K E
x : E
hx : x ∈ p.aroots E
inst✝¹ : Fintype (K →ₐ[F] E)
inst✝ : Fintype (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] 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) :=
  let_fun this := algHomEquivSigma;
  this
this : (f : K →ₐ[F] E) → Fintype (↥K⟮x⟯ →ₐ[K] E)
⊢ Fintype.card (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] E) = Fintype.card (K →ₐ[F] E) * finrank K ↥K⟮x⟯
tactic: haveI : ∀ f : K →ₐ[F] E, Fintype (@AlgHom K K⟮x⟯ E _ _ _ _ (RingHom.toAlgebra f)) := fun f => by
    have := Fintype.ofEquiv _ key_equiv
    apply Fintype.ofInjective (Sigma.mk f) fun _ _ H => eq_of_heq (Sigma.ext_iff.mp H).2
state_before: F : Type u_1
inst✝⁸ : Field F
E : Type u_2
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
p : F[X]
hFE : FiniteDimensional F E
sp : Polynomial.IsSplittingField F E p
hp : p.Separable
K : Type u_3
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra K E
inst✝² : IsScalarTower F K E
x : E
hx : x ∈ p.aroots E
inst✝¹ : Fintype (K →ₐ[F] E)
inst✝ : Fintype (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] 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) :=
  let_fun this := algHomEquivSigma;
  this
this : (f : K →ₐ[F] E) → Fintype (↥K⟮x⟯ →ₐ[K] E)
⊢ Fintype.card (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] E) = Fintype.card (K →ₐ[F] E) * finrank K ↥K⟮x⟯
state_after: F : Type u_1
inst✝⁸ : Field F
E : Type u_2
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
p : F[X]
hFE : FiniteDimensional F E
sp : Polynomial.IsSplittingField F E p
hp : p.Separable
K : Type u_3
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra K E
inst✝² : IsScalarTower F K E
x : E
hx : x ∈ p.aroots E
inst✝¹ : Fintype (K →ₐ[F] E)
inst✝ : Fintype (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] 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) :=
  let_fun this := algHomEquivSigma;
  this
this : (f : K →ₐ[F] E) → Fintype (↥K⟮x⟯ →ₐ[K] E)
⊢ ∑ i : K →ₐ[F] E, Fintype.card (↥K⟮x⟯ →ₐ[K] E) = Fintype.card (K →ₐ[F] E) * (minpoly K x).natDegree
tactic: rw [Fintype.card_congr key_equiv, Fintype.card_sigma, IntermediateField.adjoin.finrank h]
state_before: F : Type u_1
inst✝⁸ : Field F
E : Type u_2
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
p : F[X]
hFE : FiniteDimensional F E
sp : Polynomial.IsSplittingField F E p
hp : p.Separable
K : Type u_3
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra K E
inst✝² : IsScalarTower F K E
x : E
hx : x ∈ p.aroots E
inst✝¹ : Fintype (K →ₐ[F] E)
inst✝ : Fintype (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] 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) :=
  let_fun this := algHomEquivSigma;
  this
this : (f : K →ₐ[F] E) → Fintype (↥K⟮x⟯ →ₐ[K] E)
⊢ ∑ i : K →ₐ[F] E, Fintype.card (↥K⟮x⟯ →ₐ[K] E) = Fintype.card (K →ₐ[F] E) * (minpoly K x).natDegree
state_after: case h₁
F : Type u_1
inst✝⁸ : Field F
E : Type u_2
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
p : F[X]
hFE : FiniteDimensional F E
sp : Polynomial.IsSplittingField F E p
hp : p.Separable
K : Type u_3
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra K E
inst✝² : IsScalarTower F K E
x : E
hx : x ∈ p.aroots E
inst✝¹ : Fintype (K →ₐ[F] E)
inst✝ : Fintype (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] 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) :=
  let_fun this := algHomEquivSigma;
  this
this : (f : K →ₐ[F] E) → Fintype (↥K⟮x⟯ →ₐ[K] E)
⊢ ∀ x_1 ∈ Finset.univ, Fintype.card (↥K⟮x⟯ →ₐ[K] E) = (minpoly K x).natDegree
tactic: apply Finset.sum_const_nat
state_before: case h₁
F : Type u_1
inst✝⁸ : Field F
E : Type u_2
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
p : F[X]
hFE : FiniteDimensional F E
sp : Polynomial.IsSplittingField F E p
hp : p.Separable
K : Type u_3
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra K E
inst✝² : IsScalarTower F K E
x : E
hx : x ∈ p.aroots E
inst✝¹ : Fintype (K →ₐ[F] E)
inst✝ : Fintype (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] 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) :=
  let_fun this := algHomEquivSigma;
  this
this : (f : K →ₐ[F] E) → Fintype (↥K⟮x⟯ →ₐ[K] E)
⊢ ∀ x_1 ∈ Finset.univ, Fintype.card (↥K⟮x⟯ →ₐ[K] E) = (minpoly K x).natDegree
state_after: case h₁
F : Type u_1
inst✝⁸ : Field F
E : Type u_2
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
p : F[X]
hFE : FiniteDimensional F E
sp : Polynomial.IsSplittingField F E p
hp : p.Separable
K : Type u_3
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra K E
inst✝² : IsScalarTower F K E
x : E
hx : x ∈ p.aroots E
inst✝¹ : Fintype (K →ₐ[F] E)
inst✝ : Fintype (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] 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) :=
  let_fun this := algHomEquivSigma;
  this
this : (f : K →ₐ[F] E) → Fintype (↥K⟮x⟯ →ₐ[K] E)
f : K →ₐ[F] E
a✝ : f ∈ Finset.univ
⊢ Fintype.card (↥K⟮x⟯ →ₐ[K] E) = (minpoly K x).natDegree
tactic: intro f _
state_before: case h₁
F : Type u_1
inst✝⁸ : Field F
E : Type u_2
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
p : F[X]
hFE : FiniteDimensional F E
sp : Polynomial.IsSplittingField F E p
hp : p.Separable
K : Type u_3
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra K E
inst✝² : IsScalarTower F K E
x : E
hx : x ∈ p.aroots E
inst✝¹ : Fintype (K →ₐ[F] E)
inst✝ : Fintype (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] 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) :=
  let_fun this := algHomEquivSigma;
  this
this : (f : K →ₐ[F] E) → Fintype (↥K⟮x⟯ →ₐ[K] E)
f : K →ₐ[F] E
a✝ : f ∈ Finset.univ
⊢ Fintype.card (↥K⟮x⟯ →ₐ[K] E) = (minpoly K x).natDegree
state_after: case h₁
F : Type u_1
inst✝⁸ : Field F
E : Type u_2
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
p : F[X]
hFE : FiniteDimensional F E
sp : Polynomial.IsSplittingField F E p
hp : p.Separable
K : Type u_3
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra K E
inst✝² : IsScalarTower F K E
x : E
hx : x ∈ p.aroots E
inst✝¹ : Fintype (K →ₐ[F] E)
inst✝ : Fintype (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] 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) :=
  let_fun this := algHomEquivSigma;
  this
this : (f : K →ₐ[F] E) → Fintype (↥K⟮x⟯ →ₐ[K] E)
f : K →ₐ[F] E
a✝ : f ∈ Finset.univ
⊢ Fintype.card (↥K⟮x⟯ →ₐ[K] E) = Fintype.card (↥K⟮x⟯ →ₐ[K] E)

case h₁.h_sep
F : Type u_1
inst✝⁸ : Field F
E : Type u_2
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
p : F[X]
hFE : FiniteDimensional F E
sp : Polynomial.IsSplittingField F E p
hp : p.Separable
K : Type u_3
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra K E
inst✝² : IsScalarTower F K E
x : E
hx : x ∈ p.aroots E
inst✝¹ : Fintype (K →ₐ[F] E)
inst✝ : Fintype (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] 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) :=
  let_fun this := algHomEquivSigma;
  this
this : (f : K →ₐ[F] E) → Fintype (↥K⟮x⟯ →ₐ[K] E)
f : K →ₐ[F] E
a✝ : f ∈ Finset.univ
⊢ IsSeparable K x

case h₁.h_splits
F : Type u_1
inst✝⁸ : Field F
E : Type u_2
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
p : F[X]
hFE : FiniteDimensional F E
sp : Polynomial.IsSplittingField F E p
hp : p.Separable
K : Type u_3
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra K E
inst✝² : IsScalarTower F K E
x : E
hx : x ∈ p.aroots E
inst✝¹ : Fintype (K →ₐ[F] E)
inst✝ : Fintype (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] 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) :=
  let_fun this := algHomEquivSigma;
  this
this : (f : K →ₐ[F] E) → Fintype (↥K⟮x⟯ →ₐ[K] E)
f : K →ₐ[F] E
a✝ : f ∈ Finset.univ
⊢ Polynomial.Splits (algebraMap K E) (minpoly K x)
tactic: rw [← @IntermediateField.card_algHom_adjoin_integral K _ E _ _ x E _ (RingHom.toAlgebra f) h]