EPSRC DTP PhD studentship: Hopf-Galois structures on cyclic field extensions

University of Exeter - College of Engineering, Mathematics and Physical Sciences

Main supervisor: Professor Nigel Byott (University of Exeter)

Co-supervisor:  Dr Henri Johnston (University of Exeter)

This project is at the interface of abstract algebra and number theory.

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. 

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, 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.

The student's role would be to prove theorems enumerating all possible Hopf-Galois structures on a cyclic field extension, under various hypotheses on the degree of the extension. This would be done under the guidance and direction of the lead supervisor. The student would initially need to familiarise themselves thoroughly with the background to the project and with existing work on Hopf-Galois structures, and in particular with the work of myself and my current PhD student Ali Bilal on Hopf-Galois structures on cyclic extensions of square-free degree. They would then seek to extend this result by weakening the hypothesis on the degree.

Funding Minimum
3.5 year studentship: UK/EU tuition fees and an annual maintenance allowance at current Research Council rate. Current rate of £14,296 per year.

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Type / Role:

PhD

Location(s):

South West England