Divisibility conditions on the order of the reductions of algebraic numbers

SGOBBA, Pietro

doi:10.1090/mcom/3848

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Divisibility conditions on the order of the reductions of algebraic numbers

SGOBBA, Pietro

n.d. • In Mathematics of Computation

Peer Reviewed verified by ORBi

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https://hdl.handle.net/10993/46613

DOI
10.1090/mcom/3848

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Keywords :

reductions; distribution of primes; multiplicative order; Chebotarev density theorem

Abstract :

[en] Let K be a number field, and let G be a finitely generated subgroup of K*. Without relying on (GRH) we prove an asymptotic formula for the number of primes \p of K such that the order of (G mod \p) is divisible by a fixed integer. We also provide a rational expression for the natural density of this set. Furthermore, we study the primes \p for which the order is k-free, and those for which the order has a prescribed \ell-adic valuation for finitely many primes \ell. An additional condition on the Frobenius conjugacy class of \p may be considered. In order to establish these results, we prove an unconditional version of the Chebotarev density theorem for Kummer extensions of number fields.

Disciplines :

Mathematics

Author, co-author :

SGOBBA, Pietro ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)

External co-authors :

Language :

English

Title :

Divisibility conditions on the order of the reductions of algebraic numbers

Publication date :

n.d.

Journal title :

Mathematics of Computation

ISSN :

0025-5718

eISSN :

1088-6842

Publisher :

American Mathematical Society, United States

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

http://arxiv.org/abs/2110.08911

Available on ORBilu :

since 14 March 2021

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