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

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.