Nilpotent and abelian Hopf-Galois structures on field extensions

TL;DR

This paper investigates the enumeration and classification of nilpotent and abelian Hopf-Galois structures on finite Galois extensions, providing reduction techniques and criteria for cyclic extensions.

Contribution

It introduces methods to reduce the enumeration of nilpotent Hopf-Galois structures to Sylow subgroups and characterizes when abelian structures have a specific type.

Findings

Enumeration of nilpotent Hopf-Galois structures via Sylow subgroups

Conditions for abelian structures to have the same type as the Galois group

Criteria for cyclic extensions where all Hopf-Galois structures are cyclic

Abstract

Let $L/K$ be a finite Galois extension of fields with group $\Gamma$. When $\Gamma$ is nilpotent, we show that the problem of enumerating all nilpotent Hopf-Galois structures on $L/K$ can be reduced to the corresponding problem for the Sylow subgroups of $\Gamma$. We use this to enumerate all nilpotent (resp. abelian) Hopf-Galois structures on a cyclic extension of arbitrary finite degree. When $\Gamma$ is abelian, we give conditions under which every abelian Hopf-Galois structure on $L/K$ has type $\Gamma$. We also give a criterion on $n$ such that \emph{every} Hopf-Galois structure on a cyclic extension of degree $n$ has cyclic type.