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

Abstract

In this article we survey and examine the realizability of p-groups as Galois groups over arbitrary fields. In particular we consider various cohomological criteria that lead to necessary and sufficient conditions for the realizability of such a group as a Galois group, the embedding problem (i.e., realizability over a given subextension), descriptions of such extensions, automatic realizations among p-groups, and related topics.

Induced orthogonal representations of Galois groups

“…In [ST] and [Mi3] the reader can find applications of the quadratic corestriction for small 2-groups. The main result of our paper is Theorem 2.2, where we show explicitly the connection between the induced orthogonal representations and the corestriction maps.…”

Construction of unipotent Galois extensions and Massey products

“…These are cases when the existence of one Galois group over a given field forces the existence of some other Galois groups over this field. (See for example [Je,MS2,MSS,MZ,Wh].) However, nontrivial cases of automatic realizations coming from an actual construction of embedding smaller Galois extensions to larger ones, are relatively rare, and they are difficult to produce.…”

Noether's problem for some groups of order 16n

“…In Section 5 we show that for some of the groups considered in Theorem 1.2 we need only a primitive 2nth root of unity in K. To this end we apply a somewhat different approach described in Theorem 2.7. It involves calculations of the obstructions to some embedding problems, discussed recently in [Mi1,Mi2,Zi1,Zi2].…”

“…In [Mi1] and [Zi1] the reader can find two different approaches to calculating the obstructions, displayed in the following three propositions. Zi1,Th.…”