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Hopf-Galois extensions were introduced by Chase and Sweedler [CS69] in 1969, motivated by the problem of formulating an analogue of Galois theory for inseparable extensions. Their approach shed a new light on separable extensions. Later in 1987, the concept of Hopf-Galois theory was further developed by Greither and Pareigis [GP87]. So, as a problem in the theory of groups, they explained the problem of finding all Hopf-Galois structures on a finite separable extension of fields. After that, many results on Hopf-Galois structures were obtained by N. Byott, T. Crespo, S. Carnahan, L. Childs, and T. Kohl. In this thesis, we consider Hopf-Galois structures on Galois extensions of squarefree degree n. We first determine the number of isomorphism classes of groups G of order n whose centre and commutator subgroup have given orders, and we describe Aut(G) for each such G. By investigating regular cyclic subgroups in Hol(G), we enumerate the Hopf-Galois structures of type G on a cyclic extension of fields L/K of degree n. We then determine the total number of Hopf-Galois structures on L/K. Finally, we examine Hopf-Galois structures on a Galois extension L/K with arbitrary Galois group Gamma of order n, and give a formula for the number of Hopf-Galois structures on L/K of a given type G. |
