/-- If `p` is a prime that does not divide `n`,
then the minimal polynomials of a primitive `n`-th root of unity `μ`
and of `μ ^ p` are the same. -/
theorem minpoly_eq_pow {p : ℕ} [hprime : Fact p.Prime] (hdiv : ¬p ∣ n) :
    minpoly ℤ μ = minpoly ℤ (μ ^ p) := by
  classical
  by_cases hn : n = 0
  · simp_all
  have hpos := Nat.pos_of_ne_zero hn
  by_contra hdiff
  set P := minpoly ℤ μ
  set Q := minpoly ℤ (μ ^ p)
  have Pmonic : P.Monic := minpoly.monic (h.isIntegral hpos)
  have Qmonic : Q.Monic := minpoly.monic ((h.pow_of_prime hprime.1 hdiv).isIntegral hpos)
  have Pirr : Irreducible P := minpoly.irreducible (h.isIntegral hpos)
  have Qirr : Irreducible Q := minpoly.irreducible ((h.pow_of_prime hprime.1 hdiv).isIntegral hpos)
  have PQprim : IsPrimitive (P * Q) := Pmonic.isPrimitive.mul Qmonic.isPrimitive
  have prod : P * Q ∣ X ^ n - 1 := by
    rw [IsPrimitive.Int.dvd_iff_map_cast_dvd_map_cast (P * Q) (X ^ n - 1) PQprim
        (monic_X_pow_sub_C (1 : ℤ) (ne_of_gt hpos)).isPrimitive,
      Polynomial.map_mul]
    refine' IsCoprime.mul_dvd _ _ _
    · have aux := IsPrimitive.Int.irreducible_iff_irreducible_map_cast Pmonic.isPrimitive
      refine' (dvd_or_coprime _ _ (aux.1 Pirr)).resolve_left _
      rw [map_dvd_map (Int.castRingHom ℚ) Int.cast_injective Pmonic]
      intro hdiv
      refine' hdiff (eq_of_monic_of_associated Pmonic Qmonic _)
      exact associated_of_dvd_dvd hdiv (Pirr.dvd_symm Qirr hdiv)
    · apply (map_dvd_map (Int.castRingHom ℚ) Int.cast_injective Pmonic).2
      exact minpoly_dvd_x_pow_sub_one h
    · apply (map_dvd_map (Int.castRingHom ℚ) Int.cast_injective Qmonic).2
      exact minpoly_dvd_x_pow_sub_one (pow_of_prime h hprime.1 hdiv)
  replace prod := RingHom.map_dvd (mapRingHom (Int.castRingHom (ZMod p))) prod
  rw [coe_mapRingHom, Polynomial.map_mul, Polynomial.map_sub, Polynomial.map_one,
    Polynomial.map_pow, map_X] at prod
  obtain ⟨R, hR⟩ := minpoly_dvd_mod_p h hdiv
  rw [hR, ← mul_assoc, ← Polynomial.map_mul, ← sq, Polynomial.map_pow] at prod
  have habs : map (Int.castRingHom (ZMod p)) P ^ 2 ∣ map (Int.castRingHom (ZMod p)) P ^ 2 * R := by
    use R
  replace habs :=
    lt_of_lt_of_le (PartENat.coe_lt_coe.2 one_lt_two)
      (multiplicity.le_multiplicity_of_pow_dvd (dvd_trans habs prod))
  have hfree : Squarefree (X ^ n - 1 : (ZMod p)[X]) :=
    (separable_X_pow_sub_C 1 (fun h => hdiv <| (ZMod.nat_cast_zmod_eq_zero_iff_dvd n p).1 h)
        one_ne_zero).squarefree
  cases'
    (multiplicity.squarefree_iff_multiplicity_le_one (X ^ n - 1)).1 hfree
      (map (Int.castRingHom (ZMod p)) P) with
    hle hunit
  · rw [Nat.cast_one] at habs; exact hle.not_lt habs
  · replace hunit := degree_eq_zero_of_isUnit hunit
    rw [degree_map_eq_of_leadingCoeff_ne_zero (Int.castRingHom (ZMod p)) _] at hunit
    · exact (minpoly.degree_pos (isIntegral h hpos)).ne' hunit
    

Proof state:
n : ℕ
K : Type u_1
inst✝² : CommRing K
μ : K
h : IsPrimitiveRoot μ n
inst✝¹ : IsDomain K
inst✝ : CharZero K
p : ℕ
hprime : Fact (Nat.Prime p)
hdiv : ¬p ∣ n
hn : ¬n = 0
hpos : 0 < n
P : ℤ[X] := minpoly ℤ μ
Q : ℤ[X] := minpoly ℤ (μ ^ p)
hdiff : ¬P = Q
Pmonic : Monic P
Qmonic : Monic Q
Pirr : Irreducible P
Qirr : Irreducible Q
PQprim : IsPrimitive (P * Q)
R : (ZMod p)[X]
prod : map (Int.castRingHom (ZMod p)) P ^ 2 * R ∣ X ^ n - 1
hR : map (Int.castRingHom (ZMod p)) (minpoly ℤ (μ ^ p)) = map (Int.castRingHom (ZMod p)) (minpoly ℤ μ) * R
habs : ↑1 < multiplicity (map (Int.castRingHom (ZMod p)) P) (X ^ n - 1)
hfree : Squarefree (X ^ n - 1)
hunit : degree (map (Int.castRingHom (ZMod p)) P) = 0
⊢ (Int.castRingHom (ZMod p)) (leadingCoeff P) ≠ 0

Next tactic:
simp only [Pmonic, eq_intCast, Monic.leadingCoeff, Int.cast_one, Ne.def, not_false_iff,
      one_ne_zero]