Theorems · Theorem · number theory

IsPrimitiveRoot.norm_pow_sub_one_two

∀ {K : Type u} {L : Type v} [inst : Field L] {ζ : L} [inst_1 : Field K] [inst_2 : Algebra K L] {k : ℕ},
  IsPrimitiveRoot ζ (2 ^ (k + 1)) →
    ∀ [IsCyclotomicExtension {2 ^ (k + 1)} K L],
      Irreducible (Polynomial.cyclotomic (2 ^ (k + 1)) K) → (Algebra.norm K) (ζ ^ 2 ^ k - 1) = (-2) ^ 2 ^ k

If Irreducible (cyclotomic (2 ^ (k + 1)) K) (in particular for K = ℚ), then the norm of ζ ^ (2 ^ k) - 1 is (-2) ^ (2 ^ k).

Defined in

Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots

Cited by

3 results in Mathlib

Foundations

Depth 213 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldFieldAlgebraIsCyclotomicExtension