Fields Generated by Finite Rank Subgroups of $\overline{\mathbb{Q}}^*$

TL;DR

This paper proves that the field generated by the divisible hull of a finite rank subgroup of algebraic numbers has a free abelian multiplicative group, supporting a necessary condition for Rémond's generalized Lehmer conjecture.

Contribution

It establishes a structural property of fields generated by divisible hulls of finite rank subgroups, linking to Rémond's conjecture.

Findings

The multiplicative group of the generated field is free abelian modulo the divisible hull.

Supports a necessary condition for Rémond's generalized Lehmer conjecture.

Provides structural insights into fields generated by finite rank subgroups.

Abstract

Let $\Gamma$ be a finite rank subgroup of $\overline{\mathbb{Q}}^*$. We prove that the multiplicative group of the field generated by all elements in the divisible hull of $\Gamma$, is free abelian modulo this divisible hull. This proves that a necessary condition for R\'emond's generalized Lehmer conjecture is satisfied.