The order of the reductions of an algebraic integer

TL;DR

This paper investigates the distribution of primes in a number field where the reduction of a fixed algebraic integer has a multiplicative order with specific properties related to a prime l, providing explicit density calculations.

Contribution

It introduces methods to compute the density of primes with prescribed multiplicative order conditions and evaluates degrees of certain field extensions involving roots of unity and algebraic integers.

Findings

Derived explicit formulas for the density of primes with specific order properties.

Calculated degrees of extensions involving roots of unity and roots of algebraic integers.

Provided tools for understanding the distribution of primes in algebraic number fields.

Abstract

Let K be a number field, and let a be a non-zero element of K. 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_m, \sqrt[n]{a}) with n\leq 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.