On unramified solvable extensions of small number fields

Abstract: 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.