Counting Hopf-Galois structures on cyclic field extensions of squarefree degree
Alabdali, AA; Byott, NP
Date: 21 September 2017
Journal
Journal of Algebra
Publisher
Elsevier for Academic Press
Publisher DOI
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 ...
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.
Mathematics and Statistics
Faculty of Environment, Science and Economy
Item views 0
Full item downloads 0