number field; reduction; multiplicative order; density; Kummer theory
Résumé :
[en] Let \alpha_1,...,\alpha_r be algebraic numbers in a number field K generating a subgroup of rank r in K*. We investigate under GRH the number of primes p of K such that each of the orders of (\alpha_i mod p) lies in a given arithmetic progression associated to (\alpha_i). We also study the primes p for which the index of (\alpha_i mod p) is a fixed integer or lies in a given set of integers for each i. An additional condition on the Frobenius conjugacy class of p may be considered. Such results are generalizations of a theorem of Ziegler from 2006, which concerns the case r=1 of this problem.
Disciplines :
Mathématiques
Auteur, co-auteur :
SGOBBA, Pietro ; University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit
Co-auteurs externes :
no
Langue du document :
Anglais
Titre :
On the distribution of the order and index for the reductions of algebraic numbers
