Sur la Séparation des Caractères par les Frobenius

Abstract

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 ${\mathfrak p}$ of $\mathcal{O}_K$. We first recall an upper bound for the norm ${\mathrm N}({\mathfrak p})$ of the smallest such prime ${\mathfrak p}$, depending on the conductors and on the degrees. Then we give two applications: (i) find a prime number $p$ for which $P$ $(\operatorname{mod} p)$ has a certain type of factorization in ${\mathbb F}_p[X]$, where $P\in {\mathbb Z}[X]$ is a monic, irreducible polynomial of square-free discriminant; (ii) on the estimation of the maximal number of tamely ramified extensions of Galois group $A_n$ 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 $n=5,7,13$.