Galois p-groups and Galois modules

TL;DR

This paper explores the role of minimal non-abelian p-groups in Galois theory, their occurrence as Galois groups over various fields, and how their presence influences the appearance of related groups.

Contribution

It analyzes the significance of small non-abelian p-groups in Galois cohomology and characterizes their frequency and implications as Galois groups over different fields.

Findings

Minimal non-abelian p-groups are fundamental in Galois p-extensions.

Certain Galois groups' appearance implies the presence of others.

The occurrence of these groups is characterized over various base fields.

Abstract

The smallest non-abelian p-groups play a fundamental role in the theory of Galois p-extensions. We illustrate this by highlighting their role in the definition of the norm residue map in Galois cohomology. We then determine how often these groups --- as well as other closely related, larger p-groups --- occur as Galois groups over given base fields. We show further how the appearance of some Galois groups forces the appearance of other Galois groups in an interesting way.