Abelian Hopf Galois structures from almost trivial commutative nilpotent algebras

TL;DR

This paper explores specific Hopf Galois structures arising from almost trivial commutative nilpotent algebras on elementary abelian p-groups, detailing their enumeration, explicit forms, and Galois correspondence properties.

Contribution

It characterizes and counts Hopf Galois structures from abelian nilpotent algebras with minimal square, providing explicit descriptions and analyzing Galois correspondence failures.

Findings

Number of Hopf Galois structures determined

Explicit descriptions of structures provided

Extent of Galois correspondence failure estimated

Abstract

Let $L/K$ be a Galois extension of fields with Galois group $G$, an elementary abelian $p$-group of rank $n$ for $p$ an odd prime. It is known that nilpotent $\mathbb{F}_p$-algebra structures $A$ on $G$ yield regular subgroups of the holomorph of $G$, hence Hopf Galois structures on $L/K$. In this paper we illustrate the richness of Hopf Galois structures on $L/K$ by examining the case where $A$ is abelian of dimension $n$ where the dimension of $A^2 = 1$. We determine the number of Hopf Galois structures that arise in these cases, describe those structures explicitly, and estimate the extent of failure of surjectivity of the Galois correspondence for those structures.