  Bombieri and Zannier gave an effective construction of algebraic numbers of
small height inside the maximal Galois extension of the rationals which is
totally split at a given finite set of prime numbers. They proved, in
particular, an explicit upper bound for the lim inf of the height of elements
in such fields. We generalize their result in an effective way to maximal
Galois extensions of number fields with given local behaviour at finitely many
places.
