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Abstract

A subfield $K \subseteq \overline {\mathbb {Q}}$ has the Bogomolov property if there exists a positive $\varepsilon $ such that no non-torsion point of $K^\times $ has absolute logarithmic height below $\varepsilon $. We define a relative extension $L/K$ to be Bogomolov if this holds for points of $L^\times \setminus K^\times $. We construct various examples of extensions which are and are not Bogomolov. We prove a ramification criterion for this property, and use it to show that such extensions can always be constructed if some rational prime has bounded ramification index in $K$.