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We investigate Hopf-Galois structures on a cyclic field extension $L/K$ ofsquarefree degree $n$. By a result of Greither and Pareigis, each suchHopf-Galois structure corresponds to a group of order $n$, whose isomorphismclass we call the type of the Hopf-Galois structure. We show that every groupof order $n$ can occur, and we determine the number of Hopf-Galois structuresof 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, wegive closed expressions for the number of Hopf-Galois structures on a cyclicextension whose degree is a product of three distinct primes. (There areseveral cases, depending on congruence conditions between the primes.) We alsoconsider one case where the degree is a product of four primes.

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