Counting Hopf-Galois structures on cyclic field extensions of squarefree degree
Journal of Algebra
Elsevier for Academic Press
© 2017 Elsevier Inc. All rights reserved.
Reason for embargo
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.
The first-named author acknowledges support from The Higher Committee for Education Development in Iraq.
This is the author accepted manuscript. The final version is available from Elsevier via the DOI in this record.
Published online 21 September 2017