The order of the reductions of an algebraic integer

JOURNAL OF NUMBER THEORY

卷:148

Article

关键词: Number field;    Reductions;    Multiplicative order;    Primes;    Density;   

DOI  :  10.1016/j.jnt.2014.09.022

来源: Elsevier

【 摘 要 】

Let K be a number field, and let a is an element of K-x. Fix some prime number l. We compute the density of the following set: the primes p of K such that the multiplicative order of the reduction of a modulo p is coprime to l (or, more generally, has some prescribed l-adic valuation). We evaluate the degree over K of extensions of the form K((zeta(lm), l(n)root a) with n <= m, which are obtained by adjoining a root of unity of order l(m) and the l(n)-th roots of a, as this is needed for computing the above density. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free