# Computation of Hopf Galois structures on separable extensions and classification of those for degree twice an odd prime power

## TL;DR

This paper introduces a computational method to classify all Hopf Galois structures on separable extensions of finite degree, especially for degree twice an odd prime power, and provides theoretical classifications for specific degrees.

## Contribution

It presents a Magma program for classifying Hopf Galois structures and offers new theoretical results on their occurrence for degrees twice an odd prime power.

## Key findings

- A Magma program computes all Hopf Galois structures for given degrees.
- Classification of structures for degree 2p^n extensions.
- Exact determination of possible structures for degree 2p^2 extensions.

## 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 a given degree and several properties of those. We show a table which summarizes the program results. Besides, for separable field extensions of degree $2p^n$, with $p$ an odd prime number, we prove that the occurrence of some type of Hopf Galois structure may either imply or exclude the occurrence of some other type. In particular, for separable field extensions of degree $2p^2$, we determine exactly the possible sets of Hopf Galois structure types.