Reductions of subgroups of the multiplicative group

TL;DR

This paper derives a closed-form formula for the density of primes in a number field where the reduction of a finitely generated subgroup of the multiplicative group has size coprime to a fixed prime, enhancing understanding of subgroup reductions.

Contribution

It provides a new explicit formula for the density of primes with specific reduction properties of finitely generated subgroups in number fields.

Findings

Derived a closed-form expression for the prime density.

Applied the formula to various classes of subgroups.

Enhanced understanding of subgroup reduction behavior in number fields.

Abstract

Let $K$ be a number field, and let $G\subset K^\times$ be a finitely generated subgroup. Fix some prime number $\ell$, and consider the set of primes $\mathfrak{p}$ of $K$ satisfying the following property: the reduction of $G$ modulo $\mathfrak{p}$ is well-defined and has size coprime to $\ell$. We give a closed--form expression for the density of this set.