/-- If `p` is a prime and `IsCyclotomicExtension {p ^ (k + 1)} K L`, then the discriminant of
`hζ.powerBasis K` is `(-1) ^ ((p ^ (k + 1).totient) / 2) * p ^ (p ^ k * ((p - 1) * (k + 1) - 1))`
if `Irreducible (cyclotomic (p ^ (k + 1)) K))`, and `p ^ (k + 1) ≠ 2`. -/
theorem discr_prime_pow_ne_two [IsCyclotomicExtension {p ^ (k + 1)} K L] [hp : Fact (p : ℕ).Prime]
    (hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))) (hirr : Irreducible (cyclotomic (↑(p ^ (k + 1)) : ℕ) K))
    (hk : p ^ (k + 1) ≠ 2) : discr K (hζ.powerBasis K).basis =
      (-1) ^ ((p ^ (k + 1) : ℕ).totient / 2) * p ^ ((p : ℕ) ^ k * ((p - 1) * (k + 1) - 1)) := by
  haveI hne := IsCyclotomicExtension.neZero' (p ^ (k + 1)) K L
  -- Porting note: these two instances are not automatically synthesised and must be constructed
  haveI mf : Module.Finite K L := finiteDimensional {p ^ (k + 1)} K L
  haveI se : IsSeparable K L := (isGalois (p ^ (k + 1)) K L).to_isSeparable
  rw [discr_powerBasis_eq_norm, finrank L hirr, hζ.powerBasis_gen _, ←
    hζ.minpoly_eq_cyclotomic_of_irreducible hirr, PNat.pow_coe,
    totient_prime_pow hp.out (succ_pos k), Nat.add_one_sub_one]
  have coe_two : ((2 : ℕ+) : ℕ) = 2 := rfl
  have hp2 : p = 2 → k ≠ 0 := by
    rintro rfl rfl
    exact absurd rfl hk
  congr 1
  · rcases eq_or_ne p 2 with (rfl | hp2)
    · rcases Nat.exists_eq_succ_of_ne_zero (hp2 rfl) with ⟨k, rfl⟩
      rw [coe_two, succ_sub_succ_eq_sub, tsub_zero, mul_one]; simp only [_root_.pow_succ]
      rw [mul_assoc, Nat.mul_div_cancel_left _ zero_lt_two, Nat.mul_div_cancel_left _ zero_lt_two]
      cases k
      · simp
      · norm_num; simp_rw [_root_.pow_succ, (even_two.mul_right _).neg_one_pow,
          ((even_two.mul_right _).mul_right _).neg_one_pow]
    · replace hp2 : (p : ℕ) ≠ 2
      · rwa [Ne.def, ← coe_two, PNat.coe_inj]
      have hpo : Odd (p : ℕ) := hp.out.odd_of_ne_two hp2
      obtain ⟨a, ha⟩ := (hp.out.even_sub_one hp2).two_dvd
      rw [ha, mul_left_comm, mul_assoc, Nat.mul_div_cancel_left _ two_pos,
        Nat.mul_div_cancel_left _ two_pos, mul_right_comm, pow_mul, (hpo.pow.mul _).neg_one_pow,
        pow_mul, hpo.pow.neg_one_pow]
      refine' Nat.Even.sub_odd _ (even_two_mul _) odd_one
      rw [mul_left_comm, ← ha]
      exact one_le_mul (one_le_pow _ _ hp.1.pos) (succ_le_iff.2 <| tsub_pos_of_lt hp.1.one_lt)
  · have H := congr_arg (@derivative K _) (cyclotomic_prime_pow_mul_X_pow_sub_one K p k)
    rw [derivative_mul, derivative_sub, derivative_one, sub_zero, derivative_X_pow, C_eq_nat_cast,
      derivative_sub, derivative_one, sub_zero, derivative_X_pow, C_eq_nat_cast, ← PNat.pow_coe,
      hζ.minpoly_eq_cyclotomic_of_irreducible hirr] at H
    replace H := congr_arg (fun P => aeval ζ P) H
    simp only [aeval_add, aeval_mul, minpoly.aeval, zero_mul, add_zero, aeval_nat_cast,
      _root_.map_sub, aeval_one, aeval_X_pow] at H
    replace H := congr_arg (Algebra.norm K) H
    have hnorm : (norm K) (ζ ^ (p : ℕ) ^ k - 1) = (p : K) ^ (p : ℕ) ^ k := by
      by_cases hp : p = 2
      · exact mod_cast hζ.pow_sub_one_norm_prime_pow_of_ne_zero hirr le_rfl (hp2 hp)
      · exact mod_cast hζ.pow_sub_one_norm_prime_ne_two hirr le_rfl hp
    rw [MonoidHom.map_mul, hnorm, MonoidHom.map_mul, ← map_natCast (algebraMap K L),
      Algebra.norm_algebraMap, finrank L hirr] at H
    conv_rhs at H => -- Porting note: need to drill down to successfully rewrite the totient
      enter [1, 2]
      rw [PNat.pow_coe, ← succ_eq_add_one, totient_prime_pow hp.out (succ_pos k), Nat.sub_one,
        Nat.pred_succ]
    rw [← hζ.minpoly_eq_cyclotomic_of_irreducible hirr, map_pow, hζ.norm_eq_one hk hirr, one_pow,
      mul_one, PNat.pow_coe, cast_pow, ← pow_mul, ← mul_assoc, mul_comm (k + 1), mul_assoc] at H
    have := mul_pos (succ_pos k) (tsub_pos_of_lt hp.out.one_lt)
    rw [← succ_pred_eq_of_pos this, mul_succ, pow_add _ _ ((p : ℕ) ^ k)] at H
    replace H := (mul_left_inj' fun h => ?_).1 H
    · simp only [H, mul_comm _ (k + 1)]; 

Proof state:
case e_a.refine_2
p : ℕ+
k : ℕ
K : Type u
L : Type v
ζ : L
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L
hp : Fact (Nat.Prime ↑p)
hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))
hirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)
hk : p ^ (k + 1) ≠ 2
hne : NeZero ↑↑(p ^ (k + 1))
mf : Module.Finite K L
se : IsSeparable K L
coe_two : ↑2 = 2
hp2 : p = 2 → k ≠ 0
hnorm : (Algebra.norm K) (ζ ^ ↑p ^ k - 1) = ↑↑p ^ ↑p ^ k
this : 0 < succ k * (↑p - 1)
H : (Algebra.norm K) ((aeval ζ) (derivative (cyclotomic (↑p ^ (k + 1)) K))) = ↑↑p ^ (↑p ^ k * pred (succ k * (↑p - 1)))
⊢ ↑↑p ^ (↑p ^ k * pred (succ k * (↑p - 1))) = ↑↑p ^ (↑p ^ k * ((k + 1) * (↑p - 1) - 1))

Next tactic:
norm_cast