# An algorithm to determine Hopf Galois structures

## TL;DR

This paper introduces an algorithm implemented in Magma to classify all Hopf Galois structures on separable field extensions of a given degree, providing new computational tools and theoretical insights for extensions up to degree 11.

## Contribution

It presents a novel algorithm for determining all Hopf Galois structures on separable extensions and proves a uniqueness property for certain degrees.

## Key findings

- Classified Hopf Galois structures for degrees up to 11.
- Proved at most one type of structure for degrees equal to the square of an odd prime.
- Developed a Magma implementation for these classifications.

## Abstract

A Hopf Galois structure on a finite field extension L/K is given by a finite cocommutative K-Hopf algebra and a Hopf action. In this paper we present an algorithm 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 describe the results obtained for extensions of degree up to 11. Besides, we prove that separable extensions of degree equal to the square of an odd prime have at most one type of Hopf Galois structures.