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

Proof steps:
state_before: F : Type u_1
inst✝⁴ : Field F
E : Type u_2
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsGalois F E
⊢ Fintype.card (E ≃ₐ[F] E) = finrank F E
state_after: case intro
F : Type u_1
inst✝⁴ : Field F
E : Type u_2
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsGalois F E
α : E
hα : F⟮α⟯ = ⊤
⊢ Fintype.card (E ≃ₐ[F] E) = finrank F E
tactic: cases' Field.exists_primitive_element F E with α hα
state_before: case intro
F : Type u_1
inst✝⁴ : Field F
E : Type u_2
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsGalois F E
α : E
hα : F⟮α⟯ = ⊤
⊢ Fintype.card (E ≃ₐ[F] E) = finrank F E
state_after: case intro
F : Type u_1
inst✝⁴ : Field F
E : Type u_2
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsGalois F E
α : E
hα : F⟮α⟯ = ⊤
iso : ↥F⟮α⟯ ≃ₐ[F] E :=
  { toFun := fun e => ↑e, invFun := fun e => ⟨e, ⋯⟩, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯, map_add' := ⋯,
    commutes' := ⋯ }
⊢ Fintype.card (E ≃ₐ[F] E) = finrank F E
tactic: let iso : F⟮α⟯ ≃ₐ[F] E :=
    { toFun := fun e => e.val
      invFun := fun e => ⟨e, by rw [hα]; exact IntermediateField.mem_top⟩
      left_inv := fun _ => by ext; rfl
      right_inv := fun _ => rfl
      map_mul' := fun _ _ => rfl
      map_add' := fun _ _ => rfl
      commutes' := fun _ => rfl }
state_before: case intro
F : Type u_1
inst✝⁴ : Field F
E : Type u_2
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsGalois F E
α : E
hα : F⟮α⟯ = ⊤
iso : ↥F⟮α⟯ ≃ₐ[F] E :=
  { toFun := fun e => ↑e, invFun := fun e => ⟨e, ⋯⟩, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯, map_add' := ⋯,
    commutes' := ⋯ }
⊢ Fintype.card (E ≃ₐ[F] E) = finrank F E
state_after: case intro
F : Type u_1
inst✝⁴ : Field F
E : Type u_2
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsGalois F E
α : E
hα : F⟮α⟯ = ⊤
iso : ↥F⟮α⟯ ≃ₐ[F] E :=
  { toFun := fun e => ↑e, invFun := fun e => ⟨e, ⋯⟩, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯, map_add' := ⋯,
    commutes' := ⋯ }
H : IsIntegral F α
⊢ Fintype.card (E ≃ₐ[F] E) = finrank F E
tactic: have H : IsIntegral F α := IsGalois.integral F α
state_before: case intro
F : Type u_1
inst✝⁴ : Field F
E : Type u_2
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsGalois F E
α : E
hα : F⟮α⟯ = ⊤
iso : ↥F⟮α⟯ ≃ₐ[F] E :=
  { toFun := fun e => ↑e, invFun := fun e => ⟨e, ⋯⟩, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯, map_add' := ⋯,
    commutes' := ⋯ }
H : IsIntegral F α
⊢ Fintype.card (E ≃ₐ[F] E) = finrank F E
state_after: case intro
F : Type u_1
inst✝⁴ : Field F
E : Type u_2
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsGalois F E
α : E
hα : F⟮α⟯ = ⊤
iso : ↥F⟮α⟯ ≃ₐ[F] E :=
  { toFun := fun e => ↑e, invFun := fun e => ⟨e, ⋯⟩, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯, map_add' := ⋯,
    commutes' := ⋯ }
H : IsIntegral F α
h_sep : IsSeparable F α
⊢ Fintype.card (E ≃ₐ[F] E) = finrank F E
tactic: have h_sep : IsSeparable F α := IsGalois.separable F α
state_before: case intro
F : Type u_1
inst✝⁴ : Field F
E : Type u_2
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsGalois F E
α : E
hα : F⟮α⟯ = ⊤
iso : ↥F⟮α⟯ ≃ₐ[F] E :=
  { toFun := fun e => ↑e, invFun := fun e => ⟨e, ⋯⟩, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯, map_add' := ⋯,
    commutes' := ⋯ }
H : IsIntegral F α
h_sep : IsSeparable F α
⊢ Fintype.card (E ≃ₐ[F] E) = finrank F E
state_after: case intro
F : Type u_1
inst✝⁴ : Field F
E : Type u_2
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsGalois F E
α : E
hα : F⟮α⟯ = ⊤
iso : ↥F⟮α⟯ ≃ₐ[F] E :=
  { toFun := fun e => ↑e, invFun := fun e => ⟨e, ⋯⟩, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯, map_add' := ⋯,
    commutes' := ⋯ }
H : IsIntegral F α
h_sep : IsSeparable F α
h_splits : Polynomial.Splits (algebraMap F E) (minpoly F α)
⊢ Fintype.card (E ≃ₐ[F] E) = finrank F E
tactic: have h_splits : (minpoly F α).Splits (algebraMap F E) := IsGalois.splits F α
state_before: case intro
F : Type u_1
inst✝⁴ : Field F
E : Type u_2
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsGalois F E
α : E
hα : F⟮α⟯ = ⊤
iso : ↥F⟮α⟯ ≃ₐ[F] E :=
  { toFun := fun e => ↑e, invFun := fun e => ⟨e, ⋯⟩, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯, map_add' := ⋯,
    commutes' := ⋯ }
H : IsIntegral F α
h_sep : IsSeparable F α
h_splits : Polynomial.Splits (algebraMap F E) (minpoly F α)
⊢ Fintype.card (E ≃ₐ[F] E) = finrank F E
state_after: case intro
F : Type u_1
inst✝⁴ : Field F
E : Type u_2
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsGalois F E
α : E
hα : F⟮α⟯ = ⊤
iso : ↥F⟮α⟯ ≃ₐ[F] E :=
  { toFun := fun e => ↑e, invFun := fun e => ⟨e, ⋯⟩, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯, map_add' := ⋯,
    commutes' := ⋯ }
H : IsIntegral F α
h_sep : IsSeparable F α
h_splits : Polynomial.Splits (algebraMap F ↥F⟮α⟯) (minpoly F α)
⊢ Fintype.card (E ≃ₐ[F] E) = finrank F E
tactic: replace h_splits : Polynomial.Splits (algebraMap F F⟮α⟯) (minpoly F α) := by
    simpa using
      Polynomial.splits_comp_of_splits (algebraMap F E) iso.symm.toAlgHom.toRingHom h_splits
state_before: case intro
F : Type u_1
inst✝⁴ : Field F
E : Type u_2
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsGalois F E
α : E
hα : F⟮α⟯ = ⊤
iso : ↥F⟮α⟯ ≃ₐ[F] E :=
  { toFun := fun e => ↑e, invFun := fun e => ⟨e, ⋯⟩, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯, map_add' := ⋯,
    commutes' := ⋯ }
H : IsIntegral F α
h_sep : IsSeparable F α
h_splits : Polynomial.Splits (algebraMap F ↥F⟮α⟯) (minpoly F α)
⊢ Fintype.card (E ≃ₐ[F] E) = finrank F E
state_after: case intro
F : Type u_1
inst✝⁴ : Field F
E : Type u_2
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsGalois F E
α : E
hα : F⟮α⟯ = ⊤
iso : ↥F⟮α⟯ ≃ₐ[F] E :=
  { toFun := fun e => ↑e, invFun := fun e => ⟨e, ⋯⟩, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯, map_add' := ⋯,
    commutes' := ⋯ }
H : IsIntegral F α
h_sep : IsSeparable F α
h_splits : Polynomial.Splits (algebraMap F ↥F⟮α⟯) (minpoly F α)
⊢ Fintype.card (E ≃ₐ[F] E) = finrank F ↥F⟮α⟯
tactic: rw [← LinearEquiv.finrank_eq iso.toLinearEquiv]
state_before: case intro
F : Type u_1
inst✝⁴ : Field F
E : Type u_2
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsGalois F E
α : E
hα : F⟮α⟯ = ⊤
iso : ↥F⟮α⟯ ≃ₐ[F] E :=
  { toFun := fun e => ↑e, invFun := fun e => ⟨e, ⋯⟩, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯, map_add' := ⋯,
    commutes' := ⋯ }
H : IsIntegral F α
h_sep : IsSeparable F α
h_splits : Polynomial.Splits (algebraMap F ↥F⟮α⟯) (minpoly F α)
⊢ Fintype.card (E ≃ₐ[F] E) = finrank F ↥F⟮α⟯
state_after: case intro
F : Type u_1
inst✝⁴ : Field F
E : Type u_2
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsGalois F E
α : E
hα : F⟮α⟯ = ⊤
iso : ↥F⟮α⟯ ≃ₐ[F] E :=
  { toFun := fun e => ↑e, invFun := fun e => ⟨e, ⋯⟩, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯, map_add' := ⋯,
    commutes' := ⋯ }
H : IsIntegral F α
h_sep : IsSeparable F α
h_splits : Polynomial.Splits (algebraMap F ↥F⟮α⟯) (minpoly F α)
⊢ Fintype.card (E ≃ₐ[F] E) = Fintype.card (↥F⟮α⟯ ≃ₐ[F] ↥F⟮α⟯)
tactic: rw [← IntermediateField.AdjoinSimple.card_aut_eq_finrank F E H h_sep h_splits]
state_before: case intro
F : Type u_1
inst✝⁴ : Field F
E : Type u_2
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsGalois F E
α : E
hα : F⟮α⟯ = ⊤
iso : ↥F⟮α⟯ ≃ₐ[F] E :=
  { toFun := fun e => ↑e, invFun := fun e => ⟨e, ⋯⟩, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯, map_add' := ⋯,
    commutes' := ⋯ }
H : IsIntegral F α
h_sep : IsSeparable F α
h_splits : Polynomial.Splits (algebraMap F ↥F⟮α⟯) (minpoly F α)
⊢ Fintype.card (E ≃ₐ[F] E) = Fintype.card (↥F⟮α⟯ ≃ₐ[F] ↥F⟮α⟯)
state_after: case intro.f
F : Type u_1
inst✝⁴ : Field F
E : Type u_2
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsGalois F E
α : E
hα : F⟮α⟯ = ⊤
iso : ↥F⟮α⟯ ≃ₐ[F] E :=
  { toFun := fun e => ↑e, invFun := fun e => ⟨e, ⋯⟩, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯, map_add' := ⋯,
    commutes' := ⋯ }
H : IsIntegral F α
h_sep : IsSeparable F α
h_splits : Polynomial.Splits (algebraMap F ↥F⟮α⟯) (minpoly F α)
⊢ (E ≃ₐ[F] E) ≃ ↥F⟮α⟯ ≃ₐ[F] ↥F⟮α⟯
tactic: apply Fintype.card_congr
state_before: case intro.f
F : Type u_1
inst✝⁴ : Field F
E : Type u_2
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsGalois F E
α : E
hα : F⟮α⟯ = ⊤
iso : ↥F⟮α⟯ ≃ₐ[F] E :=
  { toFun := fun e => ↑e, invFun := fun e => ⟨e, ⋯⟩, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯, map_add' := ⋯,
    commutes' := ⋯ }
H : IsIntegral F α
h_sep : IsSeparable F α
h_splits : Polynomial.Splits (algebraMap F ↥F⟮α⟯) (minpoly F α)
⊢ (E ≃ₐ[F] E) ≃ ↥F⟮α⟯ ≃ₐ[F] ↥F⟮α⟯
state_after: case intro.f.left_inv
F : Type u_1
inst✝⁴ : Field F
E : Type u_2
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsGalois F E
α : E
hα : F⟮α⟯ = ⊤
iso : ↥F⟮α⟯ ≃ₐ[F] E :=
  { toFun := fun e => ↑e, invFun := fun e => ⟨e, ⋯⟩, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯, map_add' := ⋯,
    commutes' := ⋯ }
H : IsIntegral F α
h_sep : IsSeparable F α
h_splits : Polynomial.Splits (algebraMap F ↥F⟮α⟯) (minpoly F α)
⊢ Function.LeftInverse (fun ϕ => iso.symm.trans (ϕ.trans iso)) fun ϕ => iso.trans (ϕ.trans iso.symm)

case intro.f.right_inv
F : Type u_1
inst✝⁴ : Field F
E : Type u_2
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsGalois F E
α : E
hα : F⟮α⟯ = ⊤
iso : ↥F⟮α⟯ ≃ₐ[F] E :=
  { toFun := fun e => ↑e, invFun := fun e => ⟨e, ⋯⟩, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯, map_add' := ⋯,
    commutes' := ⋯ }
H : IsIntegral F α
h_sep : IsSeparable F α
h_splits : Polynomial.Splits (algebraMap F ↥F⟮α⟯) (minpoly F α)
⊢ Function.RightInverse (fun ϕ => iso.symm.trans (ϕ.trans iso)) fun ϕ => iso.trans (ϕ.trans iso.symm)
tactic: apply Equiv.mk (fun ϕ => iso.trans (ϕ.trans iso.symm)) fun ϕ => iso.symm.trans (ϕ.trans iso)