My primary research interests are in algebra and number theory. During the past fifteen years, I have been particularly interested in relative Galois module theory.

Classical Galois module theory starts with the Normal Basis Theorem for a Galois extension L of K, fields, with Galois group G, namely, that L is a free module of rank one over the group ring KG, or equivalently, that there is an element t of L so that the conjugates of t under the action of G form a basis of L as a vector space over K. If L and K are number fields, with rings of integers S, R, respectively, then the Galois group G acts on S, and a natural question is, what is the structure of S as a module over the group ring RG? The classical theorem of Emmy Noether is that S is a locally free rank one RG-module if and only if L/K is tamely ramified.

Noether's theorem is the starting point for an extensive global theory which, for a tamely ramified Galois extension L/K of algebraic number fields, relates the class of the ring of integers S of L in the locally free class group Cl(RG) (or Cl(ZG)) to analytic invariants associated with L-functions. Frohlich's book, Galois Module Structure of Algebraic Integers (Springer, 1983) gives a survey of this area of Galois module theory.

My research has been motivated by Galois module theory for Galois extensions of local fields which are wildly ramified (i.e. not tamely ramified). For wild extensions, one approach is Galois module theory is to replace the group ring RG by a larger order in KG, the associated order of S in KG, namely, the set of elements in KG which map S into itself (not just into L). This idea first arose in work of Leopoldt (1959). My interest in this area was sparked by two basic results.

One is the discovery by Greither and Pareigis (J. Algebra, vol. 106 (1987), pp. 239-258, that field extensions L/K in characteristic zero may be viewed as Hopf Galois extensions for Hopf algebras other than the group ring KG of the Galois group of L/K. In fact, if L/K is a Galois extension with Galois group G, then L/K will have at least one Hopf Galois structure other than that given by KG, unless (as N. Byott showed in 1996) G is cyclic of order n where n and (Euler's phi function)(n) are coprime.

The other is that if L/K is a Hopf Galois extension for some K-Hopf algebra A, then one can define the associated order of S in A just as above, the set of elements of A which map S to itself. It turns out that if the associated order of S in A is a Hopf order in A, then S is free of rank one as an H-module. This result of mine, a natural generalization of Noether's theorem, first appeared for H an order in A = KG, G the Galois group of L/K, in Trans. Amer. Math. Soc., vol. 304 (1987), pp. 111-140. The most general version is in a paper with D. Moss, in "Advances in Hopf Algebras", ed. by J. Bergen and S. Montgomery, Marcel Dekker, 1994, pp. 1-24.

Most of my research in recent years, and that of my students and collaborators, has been devoted to questions related to trying to understand some of the consequences of these discoveries. This research has largely centered on four problems:

- Classifying Hopf algebras (of finite rank, usually abelian) over valuation rings of local number fields;
- Understanding the principal homogeneous spaces, or equivalently, the Hopf Galois extensions, for classes of Hopf algebras;
- Classifying the Hopf Galois structures on various kinds of extensions of fields; and
- Seeking criteria for deciding if a given extension of local fields has a Hopf Galois structure for which the associated order is a Hopf order.

The last question is perhaps the ultimate goal, but in dealing with this goal it seems useful (and of intrinsic interest) to better understand the other three problems. My 1996 New York Journal of Mathematics paper, which deals with this last problem, illustrates the interplay of these questions.

My 2000 monograph, Taming Wild Extensions: Hopf Algebras and Local Galois Module Theory, (Amer. Math. Soc. Mathematical Surveys and Monographs, vol 80) gives an exposition of basic work in this area. The introduction and references for this monograph may be found elsewhere on this website.

I should also note several subsequent papers of importance in the field:

- Bondarko, M. V. Local Leopoldt's problem for rings of integers in abelian $p$-extensions of complete discrete valuation fields. Doc. Math. 5 (2000), 657--693
- Byott, Nigel P., Monogenic Hopf orders and associated orders of valuation rings, (version of July 2003) to appear in Journal of Algebra.

Byott's paper shows that a Hopf order is realizable iff it is monogenic. Bondarko's paper looks at Hopf orders in group rings of abelian groups and shows, under some restrictions, that a Hopf order is realizable iff the Hopf order represents the kernel of an isogeny of dimension one formal groups (and is then necessarily monogenic).

- Byott, Nigel P. Integral Hopf-Galois structures on degree $p^2$ extensions of $p$-adic fields. J. Algebra 248 (2002), no. 1, 334--365.

This paper extends the work of my 1996 NYJM paper to give a complete description of realizable Hopf orders in Hopf algebras of degree p^2.

- Byott, Nigel P. Hopf-Galois structures on field extensions with simple Galois groups. Bull. London Math. Soc. 36 (2004), no. 1, 23--29.

This paper extends the 1999 J. Algebra paper of Carnahan and Childs to show that if L/K is a Galois extension with a Galois group that is non-abelian and simple, then L/K has exactly two Hopf Galois structures.

During the past couple of years, my research has focused on the first and third problems above.

The first problem has led to three papers with R. Underwood. The American Journal paper adapted a construction of Hopf orders using polynomial formal groups in the 1998 AMS Memoir with Sauerberg to obtain a new class of Hopf orders ("formal group Hopf orders") inside group rings of cyclic groups of order p^n over finite extensions K of the p-adic rational numbers, and in particular, showed that Underwood's 1996 American Journal paper did not obtain all Hopf orders when n = 3. The "Duality for Hopf orders" paper, under review, constructs several different but overlapping classes of Hopf orders in KC_{p^3}, including several classes of "triangular" Hopf orders, a generalization of the construction of Underwood's paper (and Greither's 1992 Math. Z. paper in the p^2 case) and indicates how complex the landscape of Hopf orders in group rings of cyclic p-groups can be. The Illinios Journal paper, to appear, shows that the duals of some of the formal group Hopf orders in KG, G cyclic of order p^n, are triangular.

On the third problem, Steve Featherstonhaugh's 2003 doctoral thesis was a kind of uniqueness result, along the lines of Kohl's 1998 J. Algebra paper and Byott's 2004 Bull. London Math. Soc. extension of the 1999 Carnahan-Childs paper, namely that if L/K is a Galois extension of fields with Galois group G, then any H that gives a Hopf Galois structure on L/K must have group isomorphic to G. Featherstonhaugh showed that if G is an abelian p-group of rank n and exponent m, where p is sufficiently large compared to and m, then any Hopf Galois structure on L/K by an abelian K-Hopf algebra H must have that the group of H is isomorphic to G. My 2003 New York Journal paper exhibits a strong counterexample to uniqueness: if G is the holomorph of a cyclic group of order p, where p = 2q+1, q prime, then L/K has H-Hopf Galois structures where the group of H can be in any of the six isomorphism classes of groups of order |G|. Additional work on the third problem is in progress.

February 4, 2004