Theorems · Theorem · number theory

IsPrimitiveRoot.norm_eq_one

∀ {n : ℕ} [NeZero n] {K : Type u} {L : Type v} [inst : CommRing L] {ζ : L} [inst_1 : Field K] [inst_2 : Algebra K L],
  IsPrimitiveRoot ζ n →
    ∀ [IsDomain L] [IsCyclotomicExtension {n} K L],
      n ≠ 2 → Irreducible (Polynomial.cyclotomic n K) → (Algebra.norm K) ζ = 1

If Irreducible (cyclotomic n K) (in particular for K = ℚ), the norm of a primitive root is 1 if n ≠ 2.

Defined in

Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots

Cited by

3 results in Mathlib

Foundations

Depth 212 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NeZeroCommRingFieldAlgebraIsDomainIsCyclotomicExtension