  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.
