1. Counting Hopf–Galois structures on cyclic field extensions of squarefree degree
- Author
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Nigel P. Byott and Ali A. Alabdali
- Subjects
Algebra and Number Theory ,Degree (graph theory) ,Group (mathematics) ,Mathematics::Number Theory ,010102 general mathematics ,Square-free integer ,Type (model theory) ,01 natural sciences ,Combinatorics ,Field extension ,Mathematics::Quantum Algebra ,Product (mathematics) ,0103 physical sciences ,Order (group theory) ,010307 mathematical physics ,Isomorphism class ,0101 mathematics ,Mathematics - 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.
- Published
- 2018
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