Theorems · Theorem · number theory

IsPrimitiveRoot.norm_pow_sub_one_eq_prime_pow_of_ne_zero

∀ {p : ℕ} {K : Type u} {L : Type v} [inst : Field L] {ζ : L} [inst_1 : Field K] [inst_2 : Algebra K L] {k s : ℕ},
  IsPrimitiveRoot ζ (p ^ (k + 1)) →
    ∀ [hpri : Fact (Nat.Prime p)] [hcycl : IsCyclotomicExtension {p ^ (k + 1)} K L],
      Irreducible (Polynomial.cyclotomic (p ^ (k + 1)) K) →
        s ≤ k → k ≠ 0 → (Algebra.norm K) (ζ ^ p ^ s - 1) = ↑p ^ p ^ s

If Irreducible (cyclotomic (p ^ (k + 1)) K) (in particular for K = ℚ) and p is a prime, then the norm of ζ ^ (p ^ s) - 1 is p ^ (p ^ s) if k ≠ 0 and s ≤ k.

Defined in

Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots

Cited by

1 results in Mathlib

Foundations

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

Assumes

FieldFieldAlgebraFactIsCyclotomicExtension