Nigel Byott's Research Interests

My research is in algebra and algebraic number theory. It is largely concerned with Galois module structure in extensions of number fields and of local (p-adic) fields. In a tamely ramified Galois extension of number fields, the ring of integers of the top field is a locally free module over the group ring. For wildly ramified extensions, one can replace the group ring by a larger ring, the associated order. If the associated order is a Hopf order, then the ring of integers will be locally free over the associated order. Thus it is of interest to look for Hopf orders in groups algebras, and to determine when the occur as associated orders. In a slightly different direction, classical Galois theory can be generalised by considering which Hopf algebras act on a field extension. A given field extension can admit several of these Hopf-Galois structures, and one can compare the "Galois module structure" of the ring of integers in the different Hopf-Galois structures. It can happen that the ring of integers behaves "badly" in the Hopf-Galois structure coming from classical Galois theory, but "well" in some alternative Hopf-Galois structure.

My current research is concerned with the following on-going projects:

I have determined all Hopf-Galois structures on Galois extensions of fields of degree pq, where p, q are distinct primes, and hope to obtain similar results in other cases.

Given a Galois extension of number fields which is tame (i.e. at most tamely ramified), one can consider the class of the top ring of integers in the locally free classgroup of the group ring. The classes obtained from all tame extensions of a given field with a given Galois group are the realisable classes. These have been described by McCulloh for abelian extensions. We have recently described the realisable classes for the dihedral group of order 8 and the alternating group A4 of order 12. Together with Cornelius Greither, we have a result for realisable classes over the maximal order (this is weaker than working over the group ring) for a class of metabelian groups.

In certain cases, the ramification numbers for an extension of local fields are enough to determine its Galois module structure. In other cases, the ramification breaks do not give sufficient information to do so, but one can define a family of "refined" ramification breaks which contain more information. We have begun to investigate the properties of these new ramification breaks, and hope to prove that they do indeed determine the Galois module structure in some important cases.

(1) Enumerating Hopf-Galois structures on certain families of field extensions, for example dihedral groups of order a power of 2.

(2) Investigating the behaviour of rings of integers in different Hopf-Galois structures on extensions of p-adic fields with dihedral Galois group of order 2p (for an odd prime p).

(3) Determining the realisable Galois module classes for tame extensions whose Galois group is (say) a generalised quaternion group.

Here is my

Nigel Byott
School of Mathematical Sciences
University of Exeter
Exeter, EX4 4QE, UK
01392 263982
N.P.Byott@ex.ac.uk
Last updated: 18th October 2005