On the Galois groups of the 2-class towers of some imaginary quadratic fields

Let k = Q(root-2379) and k(nr,2) be the maximal unramified 2-extension of k. To show that k(n,r2)/k is finite, Michael Bush gave 8 possible presentations of finite groups for G = Gal(k(n,r2)/k). However, his methods did not further isolate G. We eliminate 4 of the possibilities, and explain how to isolate G, although carrying out the latter strategy is beyond current technological capabilities. We also discuss related examples. (c) 2006 Elsevier Inc. All rights reserved.