  We investigate unramified extensions of number fields with prescribed
solvable Galois group and certain extra conditions. In particular, we are
interested in the minimal degree of a number field $K$, Galois over
$\mathbb{Q}$, such that $K$ possesses an unramified $G$-extension. We improve
the best known bounds for the degree of such number fields $K$ for certain
classes of solvable groups, in particular nilpotent groups.
