Small points in radical extensions of number fields

We study small points in radical extensions of algebraic fields. Given an algebraic extension $\mathbb{F}$ of $\mathbb{Q}$, a finitely generated subgroup $\Gamma\subseteq \mathbb{F}^\times$, and a rational prime $p$, we give a general criterion ensuring that $\mathbb{F}(\Gamma^{p-\mathrm{div}})\setminus \Gamma^{\mathrm{div}}$ has the Bogomolov property. This problem is motivated by a conjecture of R\'emond, formulated when $\mathbb{F}$ is a number field, predicting that such radical extensions contain no unexpected small points outside the divisible hull of the group used to generate them. As applications, we obtain new cases of R\'emond's conjecture for radical extensions generated by division points with respect to a finite set of primes, recovering and extending previous results of Amoroso and the third author. Our argument is based on a recent result on small points in $p$-adic Lie extensions.