Hopf-Galois Structures on Cyclic Field Extensions

Hopf algebras provide a way to generalise classical Galois Theory. If L/K is a finite, normal, separable extension of fields, then it has a well-defined Galois group G. We can view L as a module over the group ring K[G], and the fact that the action of K[G] on L is compatible with the multiplication in L is encoded in the extra structure on K[G] which makes it into a Hopf algebra. In general, K[G] is one among many Hopf algebras H which act on L with a compatibility condition of this sort, and each of them gives L a different Hopf-Galois structure. A result of Greither and Pareigis shows that finding all possible Hopf-Galois structures on a given extension L/K amounts to a combinatorial question in group theory. Associated to each Hopf-Galois structure is a group N, of the same order as G (but not necessarily isomorphic to G). The isomorphism type of N is called the type of the Hopf-Galois structure.

For a given Galois group G, one may ask (a) how many Hopf-Galois structures of a given type N, and (b) how many Hopf-Galois structures in total, there are on Galois extension with Galois group G. These questions have been answered in some cases. For example, T. Kohl showed that if G is cyclic of order p^n for some odd prime p, then there are p^(n-1) Hopf-Galois structures, all of cyclic type. Recent work of A. Bilal and N. Byott (submitted) has determined the Hopf-Galois structures on a cyclic extension of degree m, where m is squarefree (but may have arbitrarily many distinct prime factors). This uses the fact that the groups of order m can be classified.

The aim of this PhD project is to enumerate the Hopf-Galois structures on a cyclic extension of arbitrary odd degree (or, possibly, just of arbitrary degree). Although it is no longer possible to classify all groups of order m, it follows from Kohl's result that the only relevant groups have all their Sylow subgroups cyclic (except for the prime 2), and such groups can be classified. For arbitrary m, the situation is more complicated, but it should again be possible to obtain group-theoretic restrictions on the possible types which make the problem tractable. The techniques needed will be drawn from the theory of finite groups and from elementary number theory.