Let K/ℚ be Galois, and let η∈K × whose conjugates are multiplicatively independent. For a prime p, unramified, prime to η, let n p be the residue degree of p and g p the number of 𝔭|p, then let o 𝔭 (η) and o p (η) be the orders of η modulo 𝔭 and p, respectively. Using Frobenius automorphisms, we show that for all p≫0, some explicit divisors of p n p -1 cannot realize o 𝔭 (η) nor o p (η), and we give a lower bound of o p (η). Then we obtain that, for all p≫0 such that n p >1, Prob(o p (η)<p)≤1 p g p (n p -1)-ε , where ε=O1 log 2 (p); under the Borel–Cantelli heuristic, this leads to o p (η)>p for all p≫0 such that g p (n p -1)≥2, which covers the “limit” cases of cubic fields with n p =3 and quartic fields with n p =g p =2, but not the case of quadratic fields with n p =2. In the quadratic case, the natural conjecture is, on the contrary, that o p (η)<p for infinitely many inert p. Some computations are given with PARI programs.