Hopf-Galois Structures of Squarefree Degree

TL;DR

This paper investigates the classification of Hopf-Galois structures on Galois extensions with squarefree degree, providing explicit counts and examples, especially for degrees that are products of three primes.

Contribution

It offers a complete determination of the number of Hopf-Galois structures for squarefree degrees and detailed analysis for degrees that are products of three primes.

Findings

Explicit formulas for the number of Hopf-Galois structures for squarefree degrees.

Complete classification for degrees that are the product of three primes.

Examples illustrating the theoretical results.

Abstract

Let $n$ be a squarefree natural number, and let $G$, $\Gamma$ be two groups of order $n$. We determine the number of Hopf-Galois structures of type $G$ admitted by a Galois extension of fields with Galois group isomorphic to $\Gamma$. We give some examples, including a full treatment of the case where $n$ is the product of three primes.