Theorems · Theorem · number theory

IsPrimitiveRoot.norm_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) →
    2 ≤ k →
      ∀ [H : IsCyclotomicExtension {2 ^ k} K L],
        Irreducible (Polynomial.cyclotomic (2 ^ k) K) → (Algebra.norm K) (ζ - 1) = 2

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

Defined in

Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots

Cited by

3 results in Mathlib

Foundations

Depth 215 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldFieldAlgebraIsCyclotomicExtension