Computation of Hopf Galois structures on low degree separable extensions and classification of those for degrees $p^2$ and $2p$

TL;DR

This paper develops a computational approach using Magma to classify Hopf Galois structures on low-degree separable extensions, providing new theoretical insights and explicit classifications for degrees $p^2$ and $2p$.

Contribution

It introduces a Magma program for classifying Hopf Galois structures and proves uniqueness results and classifications for degrees $p^2$ and $2p$, including explicit examples.

Findings

Classified all Hopf Galois structures on extensions up to degree 11.

Proved uniqueness of almost classically Hopf Galois structures for a given Hopf algebra.

Determined Hopf Galois structures for degrees $p^2$ and $2p$, including cyclic types.

Abstract

A Hopf Galois structure on a finite field extension $L/K$ is a pair $(H,\mu)$, where $H$ is a finite cocommutative $K$-Hopf algebra and $\mu$ a Hopf action. In this paper we present a program written in the computational algebra system Magma which gives all Hopf Galois structures on separable field extensions of degree up to eleven and several properties of those. Besides, we exhibit several results on Hopf Galois structures inspired by the program output. We prove that if $(H,\mu)$ is an almost classically Hopf Galois structure, then it is the unique Hopf Galois structure with underlying Hopf algebra $H$, up to isomorphism. For $p$ an odd prime, we prove that a separable extension of degree $p^2$ may have only one type of Hopf Galois structure and determine those of cyclic type; we determine as well the Hopf Galois structures on separable extensions of degree $2p$. We highlight the richness of the results obtained for extensions of degree 8 by computing an explicit example and presenting some tables which summarizes these results.