Galois realizability of groups of order 64
Given a field \(K\) and a group \(G\), the inverse problem of Galois theory asks whether or not there exists a Galois field extension \(L\) of \(K\) such that the Galois group of \(L\) over \(K\) is isomorphic to \(G\). In the article under review the authors determine necessary and sufficient conditions for the realizability of 134 non abelian groups of order 64 as Galois groups. The conditions are given in terms of triviality of specific elements (called \textit{obstructions}) in the Brauer group of \(K\). The obstructions are expressed as products of quaternion algebras and are written in several tables. The authors provide two sample proofs of the main types of embedding problems related to the groups. The first type produces the obstructions for 34 groups with a quotient of the type \((C_2)^r\times (C_4)^s\times (D_8)^t\). The second type produces 100 obstructions for groups known as \textit{pullbacks} (i.e., having more than one central element of order 2).
scientific article; zbMATH DE number 5900614