Realizing 2-groups as Galois groups following Shafarevich and Serre

Abstract

Let G be a finite p-group for some prime p, say of order pn. For odd p the inverse problem of Galois theory for G has been solved through the (classical) work of Scholz and Reichardt, and Serre has shown that their method leads to fields of realization where at most n rational primes are (tamely) ramified. The approach by Shafarevich, for arbitrary p, has turned out to be quite delicate in the case p=2. In this paper we treat this exceptional case in the spirit of Serre’s result, bounding the number of ramified primes at least by an integral polynomial in the rank of G, the polynomial depending on the 2-class of G.