Theorems · Theorem · number theory

IsCyclotomicExtension.norm_zeta_pow_sub_one_of_prime_ne_two

∀ {p : ℕ} {K : Type u} (L : Type v) [inst : Field K] [inst_1 : Field L] [inst_2 : Algebra K L] {k : ℕ}
  [inst_3 : Fact (Nat.Prime p)] [inst_4 : IsCyclotomicExtension {p ^ (k + 1)} K L],
  Irreducible (Polynomial.cyclotomic (p ^ (k + 1)) K) →
    p ≠ 2 → (Algebra.norm K) (IsCyclotomicExtension.zeta (p ^ (k + 1)) K L - 1) = ↑p

If Irreducible (cyclotomic (p ^ (k + 1)) K) (in particular for K = ℚ) and p is an odd prime, then the norm of zeta (p ^ (k + 1)) K L - 1 is p.

Defined in

Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots

Cited by

0 results in Mathlib

Foundations

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

Assumes

FieldFieldAlgebraFactIsCyclotomicExtension