A subfield K ⊆ ℚ̅ has the Bogomolov property if there exists a positive ε such that no non-torsion point of $K^{\times }$ has absolute logarithmic height below ε. 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.