Theorems · Theorem · number theory

IsPrimitiveRoot.not_exists_int_prime_dvd_sub_of_prime_pow_ne_two

∀ {p k : ℕ} {K : Type u} [inst : Field K] {ζ : K} [hp : Fact (Nat.Prime p)] [inst_1 : CharZero K]
  [hcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K] (hζ : IsPrimitiveRoot ζ (p ^ (k + 1))),
  p ^ (k + 1) ≠ 2 → ¬∃ n, ↑p ∣ hζ.toInteger - ↑n

In a p ^ (k + 1)-th cyclotomic extension of ℚ , we have that ζ is not congruent to an integer modulo p if p ^ (k + 1) ≠ 2.

Defined in

Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Cited by

1 results in Mathlib

Foundations

Depth 227 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldFactCharZeroIsCyclotomicExtension