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