On the order modulo p of an algebraic number

Georges Gras (1)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)

Let K/Q be Galois, and let eta in K* whose conjugates are multiplicatively independent. For a prime p, unramified, prime to eta, let np be the residue degree of p and gp the number of P I p, then let o_P(eta) and o_p(eta) be the orders of eta modulo P and p, respectively. Using Frobenius automorphisms, we show that for all p>>0, some explicit divisors of p^(np)-1 cannot realize o_P(eta) nor o_p(eta), and we give a lower bound of o_p(eta). Then we obtain that, for all p>>0 such that np >1, Prob(o_p(eta)p for all p>>0 such that gp.(np-1) ≥ 2, which covers the ``limit'' cases of cubic fields with np=3 and quartic fields with np=gp=2, but not the case of quadratic fields with np=2. In the quadratic case, the natural conjecture is, on the contrary, that o_p(eta) < p for infinitely many inert p. Some computations are given with PARI programs.

Mots clés: algebraic numbers; order modulo a prime; Frobenius automorphisms

Domaines: Théorie des nombres [math.NT]

Georges Gras. On the order modulo p of an algebraic number. 2017. ⟨hal-01225061v2⟩