Open Access
2017 Sur la Séparation des Caractères par les Frobenius
Charlotte Euvrard, Christian Maire
Publ. Mat. 61(2): 475-515 (2017). DOI: 10.5565/PUBLMAT6121706


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 $n=5,7,13$.


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Charlotte Euvrard. Christian Maire. "Sur la Séparation des Caractères par les Frobenius." Publ. Mat. 61 (2) 475 - 515, 2017.


Received: 26 October 2015; Revised: 10 October 2016; Published: 2017
First available in Project Euclid: 29 June 2017

zbMATH: 06781949
MathSciNet: MR3677869
Digital Object Identifier: 10.5565/PUBLMAT6121706

Primary: 11R21 , 11R44 , 11R45

Keywords: Chebotarev density theorem , frobenius , irreducible characters , Unramied extensions

Rights: Copyright © 2017 Universitat Autònoma de Barcelona, Departament de Matemàtiques

Vol.61 • No. 2 • 2017
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