/-- If `K` is a `p ^ k`-th cyclotomic extension of `ℚ`, then `(adjoin ℤ {ζ})` is the
integral closure of `ℤ` in `K`. -/
theorem isIntegralClosure_adjoin_singleton_of_prime_pow [hcycl : IsCyclotomicExtension {p ^ k} ℚ K]
    (hζ : IsPrimitiveRoot ζ ↑(p ^ k)) : IsIntegralClosure (adjoin ℤ ({ζ} : Set K)) ℤ K := by
  refine' ⟨Subtype.val_injective, @fun x => ⟨fun h => ⟨⟨x, _⟩, rfl⟩, _⟩⟩
  swap
  · rintro ⟨y, rfl⟩
    exact
      IsIntegral.algebraMap
        (le_integralClosure_iff_isIntegral.1
          (adjoin_le_integralClosure (hζ.isIntegral (p ^ k).pos)) _)
  let B := hζ.subOnePowerBasis ℚ
  have hint : IsIntegral ℤ B.gen := (hζ.isIntegral (p ^ k).pos).sub isIntegral_one
-- Porting note: the following `haveI` was not needed because the locale `cyclotomic` set it
-- as instances.
  letI := IsCyclotomicExtension.finiteDimensional {p ^ k} ℚ K
  have H := discr_mul_isIntegral_mem_adjoin ℚ hint h
  obtain ⟨u, n, hun⟩ := discr_prime_pow_eq_unit_mul_pow' hζ
  rw [hun] at H
  replace H := Subalgebra.smul_mem _ H u.inv
-- Porting note: the proof is slightly different because of coercions.
  rw [← smul_assoc, ← smul_mul_assoc, Units.inv_eq_val_inv, zsmul_eq_mul, ← Int.cast_mul,
    Units.inv_mul, Int.cast_one, one_mul, smul_def, map_pow] at H
  cases k
  · haveI : IsCyclotomicExtension {1} ℚ K := by simpa using hcycl
    have : x ∈ (⊥ : Subalgebra ℚ K) := by
      rw [singleton_one ℚ K]
      exact mem_top
    obtain ⟨y, rfl⟩ := mem_bot.1 this
    replace h := (isIntegral_algebraMap_iff (algebraMap ℚ K).injective).1 h
    obtain ⟨z, hz⟩ := IsIntegrallyClosed.isIntegral_iff.1 h
    rw [← hz, ← IsScalarTower.algebraMap_apply]
    exact Subalgebra.algebraMap_mem _ _
  · have hmin : (minpoly ℤ B.gen).IsEisensteinAt (Submodule.span ℤ {((p : ℕ) : ℤ)}) := by
      have h₁ := minpoly.isIntegrallyClosed_eq_field_fractions' ℚ hint
      have h₂ := hζ.minpoly_sub_one_eq_cyclotomic_comp (cyclotomic.irreducible_rat (p ^ _).pos)
      rw [IsPrimitiveRoot.subOnePowerBasis_gen] at h₁
      rw [h₁, ← map_cyclotomic_int, show Int.castRingHom ℚ = algebraMap ℤ ℚ by rfl,
        show X + 1 = map (algebraMap ℤ ℚ) (X + 1) by simp, ← map_comp] at h₂
      haveI : CharZero ℚ := StrictOrderedSemiring.to_charZero
      rw [IsPrimitiveRoot.subOnePowerBasis_gen,
        map_injective (algebraMap ℤ ℚ) (algebraMap ℤ ℚ).injective_int h₂]
      exact cyclotomic_prime_pow_comp_X_add_one_isEisensteinAt p _
    refine'
      adjoin_le _
        (mem_adjoin_of_smul_prime_pow_smul_of_minpoly_isEisensteinAt (n := n)
          (Nat.prime_iff_prime_int.1 hp.out) hint h (by simpa using H) hmin)
    

Proof state:
case refine'_1.intro.intro.succ
p : ℕ+
K : Type u
inst✝¹ : Field K
inst✝ : CharZero K
ζ : K
hp : Fact (Nat.Prime ↑p)
x : K
h : IsIntegral ℤ x
u : ℤˣ
n n✝ : ℕ
hcycl : IsCyclotomicExtension {p ^ Nat.succ n✝} ℚ K
hζ : IsPrimitiveRoot ζ ↑(p ^ Nat.succ n✝)
B : PowerBasis ℚ K := IsPrimitiveRoot.subOnePowerBasis ℚ hζ
hint : IsIntegral ℤ B.gen
this : FiniteDimensional ℚ K := finiteDimensional {p ^ Nat.succ n✝} ℚ K
hun : Algebra.discr ℚ ⇑(IsPrimitiveRoot.subOnePowerBasis ℚ hζ).basis = ↑↑u * ↑↑p ^ n
H : (algebraMap ℚ K) ↑↑p ^ n * x ∈ adjoin ℤ {B.gen}
hmin : IsEisensteinAt (minpoly ℤ B.gen) (Submodule.span ℤ {↑↑p})
⊢ {B.gen} ⊆ ↑(adjoin ℤ {ζ})

Next tactic:
simp only [Set.singleton_subset_iff, SetLike.mem_coe]