  An algebraic extension K of the rationals has the Bogomolov property if the
absolute logarithmic height of non-torsion points of K* is bounded away from 0.
We define a relative extension L/K to be Bogomolov if this holds for points of
L\K. 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, when K is Galois over Q.
