Galois realizability of groups of orders $p^5$ and $p^6$

TL;DR

This paper investigates the conditions under which groups of orders p^5 and p^6 can be realized as Galois groups over fields with characteristic not p, focusing on obstructions related to abelian quotients and cyclic algebras.

Contribution

It characterizes obstructions to Galois realizability for specific p-groups of orders p^5 and p^6, extending understanding of Galois groups in these cases.

Findings

Obstructions are expressed as products of p-cyclic algebras.

Results depend on the presence of certain roots of unity in the field.

Provides explicit criteria for Galois realizability of these groups.

Abstract

Let $p$ be an odd prime, and let $k$ be an arbitrary field of characteristic not $p$. In this article we determine the obstructions for the realizability as Galois groups over $k$ of all groups of orders $p^5$ and $p^6$, that have an abelian quotient obtained by factoring out central subgroups of order $p$ or $p^2$. These obstructions are decomposed as products of $p$-cyclic algebras, provided that $k$ contains certain roots of unity.