ON THE SEPARATION OF THE CHARACTERS BY FROBENIUS

In this paper, we are interested in the question of separating two characters of the absolute Galois group of a number field K, by the Frobenius of a prime ideal p of O-K. We first recall an upper bound for the norm N(p) of the smallest such prime p, depending on the conductors and on the degrees. Then we give two applications: (i) find a prime number p for which P (mod p) has a certain type of factorization in F-p [X], where P is an element of Z[X] is a monic, irreducible polynomial of square free discriminant; (ii) on the estimation of the maximal number of tamely ramified extensions of Galois group An over a fixed number field K. To finish, we discuss some statistics in the quadratic number fields case (real and imaginary) concerning the separation of two irreducible unramified characters of the alterning group A(n), for n = 5, 7, 13.