Computation of Galois groups associated to the 2-class towers of some quadratic fields

Abstract

The $p$-group generation algorithm from computational group theory is used to obtain information about large quotients of the pro-2 group $G=\text{Gal}(k^{nr,2}/k)$ for $k=\mathbb{Q}(\sqrt{d})$ with $d=-445,-1015,-1595,-2379$. In each case we are able to narrow the identity of $G$ down to one of a finite number of explicitly given finite groups. From this follow several results regarding the corresponding 2-class tower. This is a revised version of ANT-0302.