On realizability of $p$-groups as Galois groups

Ivo M. Michailov, Nikola P. Ziapkov

Published 2011-12-07, updated 2012-01-05Version 2

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.