Theorems · Theorem · number theory

IsPrimitiveRoot.norm_toInteger_sub_one_of_prime_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))),
  p ≠ 2 → (Algebra.norm ℤ) (hζ.toInteger - 1) = ↑p

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

Defined in

Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Cited by

5 results in Mathlib

Foundations

Depth 219 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldFactCharZeroIsCyclotomicExtension