Some infinite extensions of \(\mathbb{Q}\) satisfying Bogomolov’s property

Let \( \ell \) be a prime number and \( K_{\ell} = \mathbb{Q}(\zeta_{\ell}) \) the cyclotomic field, where \( \zeta_{\ell} \) is a primitive \( \ell \)-root of unity. Choosing a prime ideal \( \mathfrak{p} \subseteq \mathcal{O}_{K_{\ell}} \) in the ring of algebraic integers of \( K_{\ell} \), we denote by \( S_{\mathfrak{p},\ell} \) all the Galois extensions of \( K_{\ell} \) of degree \( \ell \) where \( \mathfrak{p} \) does not split. Let \( L_{\mathfrak{p},\ell} \) be the compositum of Hilbert class fields of the fields of \( S_{\mathfrak{p},\ell} \). In this work, we show that \( L_{\mathfrak{p},\ell} \) satisfies Bogomolov's property by analyzing certain local degrees over \( K_{\ell} \). We also study the relation between \( L_{\mathfrak{p},\ell} \) and other families present in the literature satisfying Bogomolov's property in the case \( \ell = 2 \).