On the distribution of the order and index for the reductions of algebraic numbers

English

Sgobba, Pietro[University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]

2020

No

[en] number field ; reduction ; multiplicative order ; density ; Kummer theory

[en] Let \alpha_1,...,\alpha_r be algebraic numbers in a number field K generating a torsion-free 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.