/-- This is the heart of the proof of the primitive element theorem. It shows that if `F` is
infinite and `α` and `β` are separable over `F` then `F⟮α, β⟯` is generated by a single element. -/
theorem primitive_element_inf_aux [IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ := by
  have hα := IsSeparable.isIntegral F α
  have hβ := IsSeparable.isIntegral F β
  let f := minpoly F α
  let g := minpoly F β
  let ιFE := algebraMap F E
  let ιEE' := algebraMap E (SplittingField (g.map ιFE))
  obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g
  let γ := α + c • β
  suffices β_in_Fγ : β ∈ F⟮γ⟯
  · use γ
    apply le_antisymm
    · rw [adjoin_le_iff]
      have α_in_Fγ : α ∈ F⟮γ⟯ := by
        rw [← add_sub_cancel α (c • β)]
        exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
      rintro x (rfl | rfl) <;> assumption
    · rw [adjoin_simple_le_iff]
      have α_in_Fαβ : α ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert α {β})
      have β_in_Fαβ : β ∈ F⟮α, β⟯ := subset_adjoin F {α, β} (Set.mem_insert_of_mem α rfl)
      exact F⟮α, β⟯.add_mem α_in_Fαβ (F⟮α, β⟯.smul_mem β_in_Fαβ)
  let p := EuclideanDomain.gcd ((f.map (algebraMap F F⟮γ⟯)).comp
    (C (AdjoinSimple.gen F γ) - (C ↑c : F⟮γ⟯[X]) * X)) (g.map (algebraMap F F⟮γ⟯))
  let h := EuclideanDomain.gcd ((f.map ιFE).comp (C γ - C (ιFE c) * X)) (g.map ιFE)
  have map_g_ne_zero : g.map ιFE ≠ 0 := map_ne_zero (minpoly.ne_zero hβ)
  have h_ne_zero : h ≠ 0 :=
    mt EuclideanDomain.gcd_eq_zero_iff.mp (not_and.mpr fun _ => map_g_ne_zero)
  suffices p_linear : p.map (algebraMap F⟮γ⟯ E) = C h.leadingCoeff * (X - C β)
  · have finale : β = algebraMap F⟮γ⟯ E (-p.coeff 0 / p.coeff 1) := by
      rw [map_div₀, RingHom.map_neg, ← coeff_map, ← coeff_map, p_linear]
      -- Porting note: had to add `-map_add` to avoid going in the wrong direction.
      simp [mul_sub, coeff_C, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero),
        -map_add]
      -- Porting note: an alternative solution is:
      -- simp_rw [Polynomial.coeff_C_mul, Polynomial.coeff_sub, mul_sub,
      --   Polynomial.coeff_X_zero, Polynomial.coeff_X_one, mul_zero, mul_one, zero_sub, neg_neg,
      --   Polynomial.coeff_C, eq_self_iff_true, Nat.one_ne_zero, if_true, if_false, mul_zero,
      --   sub_zero, mul_div_cancel_left β (mt leadingCoeff_eq_zero.mp h_ne_zero)]
    

Proof state:
case β_in_Fγ
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
ϕ : F →+* E
α β : E
inst✝¹ : Algebra F E
inst✝ : IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* SplittingField (Polynomial.map ιFE g) := algebraMap E (SplittingField (Polynomial.map ιFE g))
c : F
hc :
  ∀ α' ∈ roots (Polynomial.map (RingHom.comp ιEE' ιFE) f),
    ∀ β' ∈ roots (Polynomial.map (RingHom.comp ιEE' ιFE) g), -(α' - ιEE' α) / (β' - ιEE' β) ≠ (RingHom.comp ιEE' ιFE) c
γ : E := α + c • β
p : (↥F⟮γ⟯)[X] :=
  EuclideanDomain.gcd
    (comp (Polynomial.map (algebraMap F ↥F⟮γ⟯) f)
      (C (AdjoinSimple.gen F γ) - C { val := (algebraMap F E) c, property := (_ : (algebraMap F E) c ∈ F⟮γ⟯) } * X))
    (Polynomial.map (algebraMap F ↥F⟮γ⟯) g)
h : E[X] := EuclideanDomain.gcd (comp (Polynomial.map ιFE f) (C γ - C (ιFE c) * X)) (Polynomial.map ιFE g)
map_g_ne_zero : Polynomial.map ιFE g ≠ 0
h_ne_zero : h ≠ 0
p_linear : Polynomial.map (algebraMap (↥F⟮γ⟯) E) p = C (leadingCoeff h) * (X - C β)
finale : β = (algebraMap (↥F⟮γ⟯) E) (-coeff p 0 / coeff p 1)
⊢ β ∈ F⟮γ⟯

Next tactic:
rw [finale]