Computation of Galois groups associated to the 2-class towers of some imaginary quadratic fields with 2-class group C-2 x C-2 x C-2

摘要

We describe a method for the explicit computation of a list of possibilities for the Galois group G of an unramified 2-class tower that combines the p-group generation algorithm with algorithms from explicit class field theory. We successfully applied this method to 19 of the 36 imaginary quadratic fields of absolute discriminant less than 20,000 that have 2-class group (2, 2, 2), three negative prime discriminant factors in their discriminant, and whose 2-class towers have derived length at least 3. This is the only class of imaginary quadratic fields with 2-class group (2, 2, 2) and three negative prime discriminant factors not entirely classified by recent work of Benjamin, Lemmermeyer and Snyder. Additionally, among the 19 are all such fields whose 2-class towers, if infinite, would provide improved upper bounds for the root discriminant problem. In each case we show that these 2-class towers are finite, and in fact write down for each a short list of candidate groups for the associated Galois groups. Some of these results are unconditional, while some require the Generalized Riemann Hypothesis. (C) 2008 Elsevier Inc. All rights reserved.