2017/03/28 by Ali A. Alabdali, Alabdali, Ali A., Nigel P. Byott +1
Mathematics · #12F10 #16T05 #Advanced Topology and Set Theory #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.1703.09636
openalex publication_date 2017/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate Hopf-Galois structures on a cyclic field extension L/K of\nsquarefree degree n. By a result of Greither and Pareigis, each such\nHopf-Galois structure corresponds to a group of order n, whose isomorphism\nclass we call the type of the Hopf-Galois structure. We show that every group\nof order n can occur, and we determine the number of Hopf-Galois structures\nof each type. We then express the total number of Hopf-Galois structures on\nL/K as a sum over factorisations of n into three parts. As examples, we\ngive closed expressions for the number of Hopf-Galois structures on a cyclic\nextension whose degree is a product of three distinct primes. (There are\nseveral cases, depending on congruence conditions between the primes.) We also\nconsider one case where the degree is a product of four primes.\n