## Publicacions Matemàtiques

- Publ. Mat.
- Volume 61, Number 2 (2017), 475-515.

### Sur la Séparation des Caractères par les Frobenius

Charlotte Euvrard and Christian Maire

#### 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 $n=5,7,13$.

#### Article information

**Source**

Publ. Mat., Volume 61, Number 2 (2017), 475-515.

**Dates**

Received: 26 October 2015

Revised: 10 October 2016

First available in Project Euclid: 29 June 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.pm/1498701621

**Digital Object Identifier**

doi:10.5565/PUBLMAT6121706

**Mathematical Reviews number (MathSciNet)**

MR3677869

**Zentralblatt MATH identifier**

06781949

**Subjects**

Primary: 11R44: Distribution of prime ideals [See also 11N05] 11R21: Other number fields 11R45: Density theorems

**Keywords**

Chebotarev density theorem Frobenius Unramied extensions Irreducible characters

#### Citation

Euvrard, Charlotte; Maire, Christian. Sur la Séparation des Caractères par les Frobenius. Publ. Mat. 61 (2017), no. 2, 475--515. doi:10.5565/PUBLMAT6121706. https://projecteuclid.org/euclid.pm/1498701621