/-- Let `K` be the field of fraction of an integrally closed domain `R` and let `L` be a separable
extension of `K`, generated by an integral power basis `B` such that the minimal polynomial of
`B.gen` is Eisenstein at `p`. Given `z : L` integral over `R`, if `Q : R[X]` is such that
`aeval B.gen Q = p • z`, then `p ∣ Q.coeff 0`. -/
theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis K L}
    (hp : Prime p) (hBint : IsIntegral R B.gen) {z : L} {Q : R[X]} (hQ : aeval B.gen Q = p • z)
    (hzint : IsIntegral R z) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) : p ∣ Q.coeff 0 := by
  -- First define some abbreviations.
  letI := B.finiteDimensional
  let P := minpoly R B.gen
  obtain ⟨n, hn⟩ := Nat.exists_eq_succ_of_ne_zero B.dim_pos.ne'
  have finrank_K_L : FiniteDimensional.finrank K L = B.dim := B.finrank
  have deg_K_P : (minpoly K B.gen).natDegree = B.dim := B.natDegree_minpoly
  have deg_R_P : P.natDegree = B.dim := by
    rw [← deg_K_P, minpoly.isIntegrallyClosed_eq_field_fractions' K hBint,
      (minpoly.monic hBint).natDegree_map (algebraMap R K)]
  choose! f hf using
    hei.isWeaklyEisensteinAt.exists_mem_adjoin_mul_eq_pow_natDegree_le (minpoly.aeval R B.gen)
      (minpoly.monic hBint)
  simp only [(minpoly.monic hBint).natDegree_map, deg_R_P] at hf

  -- The Eisenstein condition shows that `p` divides `Q.coeff 0`
  -- if `p^n.succ` divides the following multiple of `Q.coeff 0^n.succ`:
  suffices
      p ^ n.succ ∣ Q.coeff 0 ^ n.succ * ((-1) ^ (n.succ * n) * (minpoly R B.gen).coeff 0 ^ n) by
    have hndiv : ¬p ^ 2 ∣ (minpoly R B.gen).coeff 0 := fun h =>
      hei.not_mem ((span_singleton_pow p 2).symm ▸ Ideal.mem_span_singleton.2 h)
    refine @Prime.dvd_of_pow_dvd_pow_mul_pow_of_square_not_dvd R _ _ _ _ n hp (?_ : _ ∣ _) hndiv
    convert (IsUnit.dvd_mul_right ⟨(-1) ^ (n.succ * n), rfl⟩).mpr this using 1
    push_cast
    ring_nf
    rw [mul_comm _ 2, pow_mul, neg_one_sq, one_pow, mul_one]

  -- We claim the quotient of `Q^n * _` by `p^n` is the following `r`:
  have aux : ∀ i ∈ (range (Q.natDegree + 1)).erase 0, B.dim ≤ i + n := by
    intro i hi
    simp only [mem_range, mem_erase] at hi
    rw [hn]
    exact le_add_pred_of_pos _ hi.1
  have hintsum :
    IsIntegral R
      (z * B.gen ^ n - ∑ x : ℕ in (range (Q.natDegree + 1)).erase 0, Q.coeff x • f (x + n)) := by
    refine (hzint.mul (hBint.pow _)).sub (.sum _ fun i hi => .smul _ ?_)
    exact adjoin_le_integralClosure hBint (hf _ (aux i hi)).1
  obtain ⟨r, hr⟩ := isIntegral_iff.1 (isIntegral_norm K hintsum)
  use r

  -- Do the computation in `K` so we can work in terms of `z` instead of `r`.
  apply IsFractionRing.injective R K
  simp only [_root_.map_mul, _root_.map_pow, _root_.map_neg, _root_.map_one]
  -- Both sides are actually norms:
  calc
    _ = norm K (Q.coeff 0 • B.gen ^ n) := ?_
    _ = norm K (p • (z * B.gen ^ n) -
          ∑ x : ℕ in (range (Q.natDegree + 1)).erase 0, p • Q.coeff x • f (x + n)) :=
        (congr_arg (norm K) (eq_sub_of_add_eq ?_))
    _ = _ := ?_
  · simp only [Algebra.smul_def, algebraMap_apply R K L, Algebra.norm_algebraMap, _root_.map_mul,
      _root_.map_pow, finrank_K_L, PowerBasis.norm_gen_eq_coeff_zero_minpoly,
      minpoly.isIntegrallyClosed_eq_field_fractions' K hBint, coeff_map, ← hn]
    

Proof state:
case h.a.calc_1
R : Type u
K : Type v
L : Type z
p : R
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra R L
inst✝⁵ : Algebra R K
inst✝⁴ : IsScalarTower R K L
inst✝³ : IsSeparable K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis K L
hp : _root_.Prime p
hBint : IsIntegral R B.gen
z : L
Q : R[X]
hQ : (aeval B.gen) Q = p • z
hzint : IsIntegral R z
hei : IsEisensteinAt (minpoly R B.gen) (Submodule.span R {p})
this : FiniteDimensional K L := finiteDimensional B
P : R[X] := minpoly R B.gen
n : ℕ
hn : B.dim = succ n
finrank_K_L : FiniteDimensional.finrank K L = B.dim
deg_K_P : natDegree (minpoly K B.gen) = B.dim
deg_R_P : natDegree P = B.dim
f : ℕ → L
hf : ∀ (i : ℕ), B.dim ≤ i → f i ∈ adjoin R {B.gen} ∧ (algebraMap R L) p * f i = B.gen ^ i
aux : ∀ i ∈ Finset.erase (range (natDegree Q + 1)) 0, B.dim ≤ i + n
hintsum : IsIntegral R (z * B.gen ^ n - ∑ x in Finset.erase (range (natDegree Q + 1)) 0, coeff Q x • f (x + n))
r : R
hr :
  (algebraMap R K) r =
    (Algebra.norm K) (z * B.gen ^ n - ∑ x in Finset.erase (range (natDegree Q + 1)) 0, coeff Q x • f (x + n))
⊢ (algebraMap R K) (coeff Q 0) ^ B.dim * ((-1) ^ (B.dim * n) * (algebraMap R K) (coeff (minpoly R B.gen) 0) ^ n) =
    (algebraMap R K) (coeff Q 0) ^ B.dim * ((-1) ^ B.dim * (algebraMap R K) (coeff (minpoly R B.gen) 0)) ^ n

Next tactic:
ring