On the order modulo p of an algebraic number (for p large enough) (1511.01755v2)

Abstract: 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\textgreater{}\textgreater{}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\textgreater{}\textgreater{}0 such that np \textgreater{}1, Prob(o_p(eta)\textless{}p) $\le$ 1/p^gp.(np-1-epsilon)), where epsilon = O(1/(log_2(p))); under the Borel--Cantelli heuristic, this leads to o_p(eta)\textgreater{}p for all p\textgreater{}\textgreater{}0 such that gp.(np-1) $\ge$ 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) \textless{} p for infinitely many inert p. Some computations are given with PARI programs.