On realizability of $p$-groups as Galois groups

TL;DR

This paper surveys the conditions under which $p$-groups can be realized as Galois groups over arbitrary fields, focusing on cohomological criteria, embedding problems, and automatic realizations.

Contribution

It provides a comprehensive overview of cohomological and structural criteria for the realizability of $p$-groups as Galois groups, including new insights into embedding problems and automatic realizations.

Findings

Cohomological criteria for realizability established

Necessary and sufficient conditions identified

Descriptions of Galois extensions with $p$-groups provided

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.