On Galois cohomology and realizability of 2-groups as Galois groups
The paper under review continues the presentation of the author's research on the problem of finding whether a given finite \(p\)-group \(G\) is realizable as a Galois group over an arbitrary field \(F\) of characteristic different from \(p\) (the solution to the same problem in characteristic \(p\) is well-known consequence of the projectivity of the Galois group of the maximal \(p\)-extension). The author concentrates on the case where \(p = 2\), \(G\) is of order \(2 ^ n\), for an integer \(n \geq 4\), and \(G\) possesses a cyclic group of order \(2 ^ {n-2}\). Assuming also that \(F\) contains a primitive root of unity of degree \(2 ^ {n-3}\), \(G\) is nonabelian and \(G\) does not decompose into a proper group product, he finds necessary and sufficient conditions for realizability of \(G\) over \(F\), when \(G\) lies in \(14\) of \(22\) possible isomorphism classes. The proof of the stated result relies on \textit{Y. Ninomiya}'s classification of nonabelian finite groups of order \(p ^ n\) and exponent \(p ^ {n-2}\) (see [Math. J. Okayama Univ. 36, 1--21 (1994; Zbl 0838.20017)]). It is obtained by considering several special cases. They are settled by applying some criteria for solving embedding problems involving \(p\)-groups (one of them obtained by \textit{I. Kiming} [Can J. Math. 42, No. 5, 825--855 (1990; Zbl 0725.12004)]), which are of independent interest. This is combined with a method (worked out in the paper), applicable to \(2\)-groups, that is based on the corestriction mapping of Brauer groups.
scientific article; zbMATH DE number 5900628
scientific article