Realizability and automatic realizability of Galois groups of order 32 (Q707984)

The conditions for the realizability of the groups of orders \(4\) and \(8\) as Galois groups over arbitrary fields with characteristic different from \(2\) have been known for a long time. It was not until 1995 when \textit{A. Ledet} [``On 2-groups as Galois groups'', Can. J. Math. 47, No. 6, 1253--1273 (1995; Zbl 0849.12006)] found the decomposition of the obstructions as products of quaternion algebras of all non abelian groups of order \(16\). In 2004 \textit{[H. G. Grundman} and \textit{G. L. Stewart}, [``Galois realizability of non-split group extensions of \(C_2\) by \((C_2)^r\times (C_4)^s\times (D_4)^t\)'', J. Algebra 272, No. 2, 425--434 (2004; Zbl 1043.12004)] found the obstructions to the realizability of \(13\) groups of order \(32\) that have a quotient group isomorphic to a direct product of cyclic groups of order \(\leq 4\) and/or the dihedral group of order \(8\). A systematic study of the embedding problems concerning all non abelian groups of order \(32\) was published by the reviewer [``Groups of order 32 as Galois groups'', Serdica Math. J. 33, No. 1, 1--34 (2007; Zbl 1199.12007)]. The article under review investigates the realizability of all \(51\) groups of order \(32\) as Galois groups over an arbitrary field \(K\) of characteristic not \(2\). The authors present a comprehensive list of necessary and sufficient conditions in terms of triviality of specific elements (called \textit{obstructions}) in the Brauer group of \(K\). This list of criteria provides a unified presentation of the results obtained by many researchers in various articles. With the aid of these criteria the authors prove a wide range of automatic realizability results, wherein the realizability of one group as a Galois group over \(K\) implies the realizability of another group.