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 $p^n$. 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 $G$.