Counting Hopf-Galois structures on cyclic field extensions of squarefree degree

JOURNAL OF ALGEBRA

卷:493

Article

Alabdali, Ali A.^1,2  Byott, Nigel P.¹ 

[1] Univ Exeter, Dept Math, Coll Engn Math & Phys Sci, Exeter EX4 4QF, Devon, England

[2] Univ Mosul, Dept Math, Coll Educ Pure Sci, Mosul, Iraq

关键词: Hopf-Galois structures;    Field extensions;    Groups of squarefree order;   

DOI  :  10.1016/j.jalgebra.2017.09.009

来源: Elsevier

【 摘 要 】

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. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free