# Counting Hopf-Galois Structures on Cyclic Field Extensions of Squarefree Degree

## TL;DR

This paper classifies and counts all Hopf-Galois structures on cyclic field extensions of squarefree degree, providing explicit formulas and examples for degrees with prime factorizations.

## Contribution

It proves that every group of squarefree order can occur as a Hopf-Galois structure and derives formulas for counting structures based on prime factorization.

## Key findings

- Every group of order n occurs as a Hopf-Galois structure.
- Number of structures is expressed as a sum over factorizations of n.
- Explicit formulas are provided for degrees with three or four prime factors.

## Abstract

We investigate Hopf-Galois structures on a cyclic field extension $L/K$ of squarefree degree $n$. By a result of Greither and Pareigis, each such Hopf-Galois structure corresponds to a group of order $n$, whose isomorphism class we call the type of the Hopf-Galois structure. We show that every group of order $n$ can occur, and we determine the number of Hopf-Galois structures of each type. We then express the total number of Hopf-Galois structures on $L/K$ as a sum over factorisations of $n$ into three parts. As examples, we give closed expressions for the number of Hopf-Galois structures on a cyclic extension whose degree is a product of three distinct primes. (There are several cases, depending on congruence conditions between the primes.) We also consider one case where the degree is a product of four primes.