Realizability and automatic realizability of Galois groups of order 32

Abstract

This article provides necessary and sufficient conditions for each group of order 32 to be realizable as a Galois group over an arbitrary field. These conditions, given in terms of the number of square classes of the field and the triviality of specific elements in related Brauer groups, are used to derive a variety of automatic realizability results.

Automatic realizations of Galois groups with cyclic quotient of order {p^n}

“…(See [4,5,6] for a good overview of the theory of automatic realizations. Some interesting automatic realizations of groups of order 16 are obtained in [2], and these and other automatic realizations of finite 2-groups are collected in [3]. For comprehensive treatments of related Galois embedding problems, see [7] and [9].…”

On Galois cohomology and realizability of 2-groups as Galois groups

“…fields with a given level or a number of square classes). There are as well a number of non trivial automatic realizations among groups of order 32 considered in [Mi4,GS3,Sm]. More automatic realizations among 2-groups can be found in three papers written by Jensen [Je1,Je2,Je3] and in a paper written by Gao, Leep, Minác and Smith [GLMS].…”

Galois realizability of a central C4-extension of D8

“…A compilation of conditions for the realizability of groups of order 2, 4, 8, or 16, as well as descriptions of extensions with those Galois groups, can be found in [5] and [8]. A compilation of conditions for all groups of order 32, based on the work in this paper as well as in [2,6,[10][11][12], can be found in [4].…”