/-- If `Irreducible (cyclotomic (p ^ (k + 1)) K)` (in particular for `K = ℚ`) and `p` is a prime,
then the norm of `ζ ^ (p ^ s) - 1` is `p ^ (p ^ s)` if `p ^ (k - s + 1) ≠ 2`. See the next lemmas
for similar results. -/
theorem pow_sub_one_norm_prime_pow_ne_two {k s : ℕ} (hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1)))
    [hpri : Fact (p : ℕ).Prime] [IsCyclotomicExtension {p ^ (k + 1)} K L]
    (hirr : Irreducible (cyclotomic (↑(p ^ (k + 1)) : ℕ) K)) (hs : s ≤ k)
    (htwo : p ^ (k - s + 1) ≠ 2) : norm K (ζ ^ (p : ℕ) ^ s - 1) = (p : K) ^ (p : ℕ) ^ s := by
-- Porting note: `by simp` was `by linarith` that now fails.
  have hirr₁ : Irreducible (cyclotomic ((p : ℕ) ^ (k - s + 1)) K) :=
    cyclotomic_irreducible_pow_of_irreducible_pow hpri.1 (by simp) hirr
  rw [← PNat.pow_coe] at hirr₁
  set η := ζ ^ (p : ℕ) ^ s - 1 with ηdef
  let η₁ : K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η
  have hη : IsPrimitiveRoot (η + 1) ((p : ℕ) ^ (k + 1 - s)) := by
    rw [sub_add_cancel]
    refine' IsPrimitiveRoot.pow (p ^ (k + 1)).pos hζ _
    rw [PNat.pow_coe, ← pow_add, add_comm s, Nat.sub_add_cancel (le_trans hs (Nat.le_succ k))]
  have : IsCyclotomicExtension {p ^ (k - s + 1)} K K⟮η⟯ := by
    suffices IsCyclotomicExtension {p ^ (k - s + 1)} K K⟮η + 1⟯.toSubalgebra by
      have H : K⟮η + 1⟯.toSubalgebra = K⟮η⟯.toSubalgebra := by
        simp only [IntermediateField.adjoin_simple_toSubalgebra_of_integral
            (IsCyclotomicExtension.integral {p ^ (k + 1)} K L _)]
        refine' Subalgebra.ext fun x => ⟨fun hx => adjoin_le _ hx, fun hx => adjoin_le _ hx⟩
        · simp only [Set.singleton_subset_iff, SetLike.mem_coe]
          exact Subalgebra.add_mem _ (subset_adjoin (mem_singleton η)) (Subalgebra.one_mem _)
-- Porting note `ηdef` was not needed.
        · simp only [Set.singleton_subset_iff, SetLike.mem_coe, ← ηdef]
          nth_rw 1 [← add_sub_cancel η 1]
          refine' Subalgebra.sub_mem _ (subset_adjoin (mem_singleton _)) (Subalgebra.one_mem _)
-- Porting note: the previous proof was `rw [H] at this; exact this` but it now fails.
      exact IsCyclotomicExtension.equiv _ _ _ (Subalgebra.equivOfEq _ _ H)
-- Porting note: the next `refine` was `rw [H]`, abusing defeq, and it now fails.
    have H := IntermediateField.adjoin_simple_toSubalgebra_of_integral
        (IsCyclotomicExtension.integral {p ^ (k + 1)} K L (η + 1))
    refine @IsCyclotomicExtension.equiv _ _ _ _ _ _ _ _ _ ?_ (Subalgebra.equivOfEq _ _ H).symm
    have hη' : IsPrimitiveRoot (η + 1) ↑(p ^ (k + 1 - s)) := by simpa using hη
-- Porting note: `using 1` was not needed.
    convert hη'.adjoin_isCyclotomicExtension K using 1
    rw [Nat.sub_add_comm hs]
  replace hη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))
  · apply coe_submonoidClass_iff.1
    convert hη using 1
    rw [Nat.sub_add_comm hs, pow_coe]
-- Porting note: the following `haveI` were not needed because the locale `cyclotomic` set them
-- as instances.
  haveI := IsCyclotomicExtension.finiteDimensional {p ^ (k + 1)} K L
  haveI := IsCyclotomicExtension.isGalois (p ^ (k + 1)) K L
  rw [norm_eq_norm_adjoin K]
  · have H := hη.sub_one_norm_isPrimePow ?_ hirr₁ htwo
    swap; · rw [PNat.pow_coe]; 

Proof state:
case refine_1
p n : ℕ+
A : Type w
B : Type z
K : Type u
L : Type v
C : Type w
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : IsCyclotomicExtension {n} A B
inst✝³ : Field L
ζ : L
hζ✝ : IsPrimitiveRoot ζ ↑n
inst✝² : Field K
inst✝¹ : Algebra K L
k s : ℕ
hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))
hpri : Fact (Nat.Prime ↑p)
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L
hirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)
hs : s ≤ k
htwo : p ^ (k - s + 1) ≠ 2
hirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)
η : L := ζ ^ ↑p ^ s - 1
ηdef : η = ζ ^ ↑p ^ s - 1
η₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η
this✝¹ : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯
hη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))
this✝ : FiniteDimensional K L
this : IsGalois K L
⊢ IsPrimePow (↑p ^ (k - s + 1))

Next tactic:
exact hpri.1.isPrimePow.pow (Nat.succ_ne_zero _)