Counting Hopf–Galois structures on cyclic field extensions of squarefree degree
Counting Hopf–Galois structures on cyclic field extensions of squarefree degree
复制标题
计算无平方度循环场扩展上的 Hopf-Galois 结构
DOI：
10.1016/j.jalgebra.2017.09.009
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发表时间：
2017
影响因子：
0.9
通讯作者：
N. Byott
中科院分区：
文献类型：
--
作者：
Ali A. Alabdali;N. Byott
摘要 我们研究了无平方度 n 的循环场扩展 L/K 上的 Hopf-Galois 结构。根据 Greither 和 Pareigis 的结果，每个这样的 Hopf-Galois 结构都对应于一个 n 阶群，我们将其同构类称为 Hopf-Galois 结构的类型。我们证明了每一个 n 阶群都可以出现，并且我们确定了每种类型的 Hopf-Galois 结构的数量。然后，我们将 L/K 上的 Hopf-Galois 结构总数表示为 n 分解为三部分的总和。作为例子，我们给出了循环扩展上的 Hopf-Galois 结构的数量的闭合表达式，其次数是三个不同素数的乘积。（有几种情况，取决于素数之间的同余条件。）我们还考虑一种情况，其中次数是四个素数的乘积。
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.