On galois groups and their maximal 2-subgroups

Abstract

LetG be a finite group of even order, having a central element of order 2 which we denote by −1. IfG is a 2-group, letG be a maximal subgroup ofG containing −1, otherwise letG be a 2-Sylow subgroup ofG. LetH=G/{±1} andH=G/{±1}. Suppose there exists a regular extensionL 1 of ℚ(T) with Galois groupG. LetL be the subfield ofL 1 fixed byH. We make the hypothesis thatL 1 admits a quadratic extensionL 2 which is Galois overL of Galois groupG. IfG is not a 2-group we show thatL 1 then admits a quadratic extension which is Galois over ℚ(T) of Galois groupG and which can be given explicitly in terms ofL 2. IfG is a 2-group, we show that there exists an element α e ℚ(T) such thatL 1 admits a quadratic extension which is Galois over ℚ(T) of Galois groupG if and only if the cyclic algebra (L/ℚ(T).a) splits. As an application of these results we explicitly construct several 2-groups as Galois groups of regular extensions of ℚ(T).