/-- 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)]
    rw [finale]
    exact Subtype.mem (-p.coeff 0 / p.coeff 1)
  have h_sep : h.Separable := separable_gcd_right _ (IsSeparable.separable F β).map
  have h_root : h.eval β = 0 := by
    apply eval_gcd_eq_zero
    · rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map, ← aeval_def, ←
        Algebra.smul_def, add_sub_cancel, minpoly.aeval]
    · rw [eval_map, ← aeval_def, minpoly.aeval]
  have h_splits : Splits ιEE' h :=
    splits_of_splits_gcd_right ιEE' map_g_ne_zero (SplittingField.splits _)
  have h_roots : ∀ x ∈ (h.map ιEE').roots, x = ιEE' β := by
    intro x hx
    rw [mem_roots_map h_ne_zero] at hx
    specialize hc (ιEE' γ - ιEE' (ιFE c) * x) (by
      have f_root := root_left_of_root_gcd hx
      rw [eval₂_comp, eval₂_sub, eval₂_mul, eval₂_C, eval₂_C, eval₂_X, eval₂_map] at f_root
      exact (mem_roots_map (minpoly.ne_zero hα)).mpr f_root)
    specialize hc x (by
      rw [mem_roots_map (minpoly.ne_zero hβ), ← eval₂_map]
      exact root_right_of_root_gcd hx)
    by_contra a
    apply hc
    apply (div_eq_iff (sub_ne_zero.mpr a)).mpr
    simp only [Algebra.smul_def, RingHom.map_add, RingHom.map_mul, RingHom.comp_apply]
    ring
  

Proof state:
case p_linear
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
h_sep : Separable h
h_root : eval β h = 0
h_splits : Splits ιEE' h
h_roots : ∀ x ∈ roots (Polynomial.map ιEE' h), x = ιEE' β
⊢ Polynomial.map (algebraMap (↥F⟮γ⟯) E) p = C (leadingCoeff h) * (X - C β)

Next tactic:
rw [← eq_X_sub_C_of_separable_of_root_eq h_sep h_root h_splits h_roots]