Theorems · Theorem · number theory

IsPrimitiveRoot.norm_toInteger_pow_sub_one_of_prime_pow_ne_two

∀ {p k : ℕ} {K : Type u} [inst : Field K] {ζ : K} [hp : Fact (Nat.Prime p)] [inst_1 : CharZero K]
  [IsCyclotomicExtension {p ^ (k + 1)} ℚ K] (hζ : IsPrimitiveRoot ζ (p ^ (k + 1))) {s : ℕ},
  s ≤ k → p ^ (k - s + 1) ≠ 2 → (Algebra.norm ℤ) (hζ.toInteger ^ p ^ s - 1) = ↑p ^ p ^ s

The norm, relative to ℤ, of ζ ^ p ^ s - 1 in a p ^ (k + 1)-th cyclotomic extension of ℚ is p ^ p ^ s if s ≤ k and p ^ (k - s + 1) ≠ 2.

Defined in

Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Cited by

3 results in Mathlib

Foundations

Depth 217 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldFactCharZeroIsCyclotomicExtension