Galois module theory is the branch of algebraic number theory which studies rings of integers of Galois extensions of number fields as modules over the integral group ring of the Galois group.

The title of this book comes from {Ch87}, which suggested that Hopf algebras could fruitfully broaden the domain of Galois module theory. That suggestion has led to a body of research towards understanding how Hopf algebras relate to wildly ramified extensions of local fields. The purpose of this book is to survey this work.

We begin by reviewing the classical theory.

Let L/K be a Galois extension of algebraic number fields with Galois group G. The normal basis theorem, dating back to the 19th century, is that L has a K-basis consisting of {\sigma(s) | \sigma in G }, the conjugates of some element s of L--a normal basis. Thus as a module over the group ring KG, L is a free module of rank one with basis s.

Let R be the ring of integers of K (or the valuation ring of K, if K is a local field), and let S be the integral closure of R in L. Then the Galois group G acts on S, and one can ask if S has a normal basis as a free R-module (alternate terminology: L/K has a normal integral basis). Of course the first prerequisite is that S be a free R-module. In the local case, i.e. if K is a local field, this holds, but in the global case S need not be free over R.

The two classic theorems in Galois module theory are the Hilbert-Speiser Theorem, Satz 132 of Hilbert's Zahlbericht {Hi97}, and Noether's Theorem {No32}. Noether's theorem is that in the local case, S has a normal basis over R if and only if L/K is tamely ramified (or, for short, " tame" ): if e_{L/K} is the ramification index of the extension L/K, then L/K is tamely ramified if e_{L/K} is relatively prime to the characteristic of the residue field of R. Two equivalent characterizations of tameness are:

1) the trace map from S to R, tr(s) = \sum_{\sigma in G} \sigma(s), is onto,

and

2) the natural action of the group ring RG on S makes S into a projective RG-module.

The difficult part of Noether's theorem is that if R is the valuation ring of a local field K, and M is a finitely generated, projective RG-module so that K tensor_R M is KG-free, then M is RG-free. Thus the first really satisfactory proof of Noether's theorem was given only in 1960 by R. G. Swan {Sw60}.

The Hilbert-Speiser Theorem asserts that if K = \Bbb Q and L/K is any tamely ramified abelian extension, then the ring of integers S of L has a normal basis as a free \Bbb Z-module, or equivalently, S is a free \Bbb Z G-module of rank one. (It was recently proven {GRRS 99} that \Bbb Q is the only base field K over which every tame abelian extension has a normal integral basis.)

The case of wild(ly ramified) extensions remained untouched by these results. Since wild extensions include all ramified Galois extensions of a local field K containing \Bbb Q_p where the Galois group is a p-group, this was a substantial omission.

The first new approach to wild extensions was by Leopoldt {Le59}, who proposed replacing the group ring RG by a larger order. More precisely, let L/K be a Galois extension of local or global number fields, with rings of integers (or valuation rings) S/R and Galois group G. Define the associated order of S in KG to be \frak A = {\alpha in KG | \alpha(s) in S for all s in S }. Since G acts on S (elements of G send elements integral over R to elements integral over R), RG \subset \frak A, but if L/K is wildly ramified, then \frak A will be larger than RG, but still an order over R in KG: that is, a subring of KG which is a finitely generated R-module.

If L/K is tame, then \frak A = RG.

Leopoldt obtained a very satisfying generalization of the Hilbert-Speiser theorem, namely, if L/\Bbb Q is any abelian extension, then the ring of integers S of L is a free \frak A-module of rank one. This implies the corresponding result for abelian extensions of \Bbb Q_p.

However, it was soon found that if one varies the hypotheses: replace \Bbb Q or \Bbb Q_p by a more general base field, or allow G to be non-abelian, then even the local result is not valid. (However for extensions L/K with L/\Bbb Q_p abelian, the local result remains true {Lt98}.)

The observation of {Ch87} for abelian extensions and {CM94} in general, was that for wild Galois extensions, if the associated order \frak A is a Hopf order in KG, then S is free of rank one over \frak A. Noether's theorem is the case where \frak A = RG. This generalized Noether's theorem was perhaps the first general integral normal basis theorem for wildly ramified Galois extensions of local fields. (The converse is false: see (12.8).)

At about the same time, Greither and Pareigis {GP87} showed how ubiquitous Hopf Galois structures were on Galois, and even separable but non-Galois, extensions of local or global fields. Their observation expands the horizon greatly. For suppose L/K is a H-Hopf Galois extension of number fields, for some cocommutative K-Hopf algebra H (the terminology means that L is an H-module algebra). If S is the ring of integers of L, one can define the associated order of S in H in the same way as for H = KG: namely,

\frak A = {\alpha in H | \alpha(s) in S for all s in S }. Then the same result holds: if \frak A is a Hopf order in H, then S is free of rank one over \frak A.

If the Galois group G of a Galois extension L/K is not cyclic of square-free order, then L/K has Hopf Galois structures other that given by the Galois group KG. For each Hopf algebra action H tensor L \to L which makes L/K into a Hopf Galois extension, there is an associated order \frak A_H. Which is the "best" order? It need not be \frak A_{KG}. N. Byott {By99b} has given a class of examples of Galois extensions of local fields for which the ring of integers S of L is not free over \frak A_{KG} but \frak A_H is a Hopf order for some other K-Hopf algebra H, and hence S is free over \frak A_H: his examples, presented in section 40, conclude this work.

Chapter Summary

The main purpose of this book is to present mathematics related to the generalized Noether theorem cited above. Here is a brief description of the twelve chapters of this work.

Chapter 1 is an introduction to the basic concepts of Hopf algebras and Hopf Galois extensions. It includes an extended discussion of exact sequences of Hopf algebras, adapted from {Sw69} and {Mt93}, and some results on Hopf orders and their integrals in section 5.

Chapter 2 surveys the Greither-Pareigis {GP87} classification of Hopf Galois structures on separable field extensions. Their classification transforms the problem of classifying Hopf Galois structures on a Galois extension into a purely group-theoretic problem involving the Galois group G, which, however, is often unmanageable. A reformulation by Byott {By96a} breaks up the problem into a possibly unmanageable collection of manageable problems. We then give Byott's classification of Galois extensions for which the classical Galois structure is the unique Hopf Galois structure, and survey results on the number of Hopf Galois structures on Galois extensions with Galois group G for various G, including cyclic p-groups {Ko98}, and symmetric, alternating and simple groups {CC99}.

Associated orders and the concept of tame H-extension, from {CH86}, are considered in Chapter 3. Once one proves that an extension S/R is H-tame iff S is H-projective, then Noether's theorem comes down to showing that if A is a Hopf order over the valuation ring R of a local field K in some cocommutative K-Hopf algebra H, and P and Q are finitely generated projective left A-modules such that K tensor_R P is isomorphic to K tensor_R Q, then P is isomorphic to Q. This result, found for H = KG in {Sw60} and also in {CR81}, and proved in general by Schneider {Sch77}, requires a very substantial amount of representation theory, so the proof is omitted. We include a proof, due to Waterhouse {Wa92} that if H is local and S/R is H-tame, then S/R is H- Galois.

In attempting to apply the generalized Noether theorem it has been helpful to understand the possible Hopf algebras over the valuation ring of a local field K which can be associated orders of valuation rings of extensions of K. Chapter 4, together with chapters 5 and 9, describe some families of Hopf orders over valuation rings of local fields. Chapter 4 presents the classification of Hopf algebras of rank p over valuation rings of local fields, due to Tate and Oort {TO70}. Mazur {Mz70} noted that describing all the possible abelian group schemes over the valuation ring of a local field "is a delicate matter", citing {TO70} as evidence. His remark remains valid 30 years later.

Chapter 5 presents Larson's construction {La76} of Hopf algebra orders in group rings which are defined by group valuations. Larson's construction remains the only general strategy for constructing Hopf orders in group rings of non-abelian groups. Section 20 obtains a class of examples of K-Hopf algebras which contain no Hopf orders at all.

Chapter 6 considers cyclic Galois extensions L/K of degree p, prime, of local number fields containing \Bbb Q_p {Ch87}, {Gr92}. In this case, a congruence condition on the ramification number, or break number, characterizes when the associated order of the valuation ring of L is a Hopf order.

Chapter 7, from {Ch88}, considers L/K an extension of number fields, and briefly discusses orders over R in L, other than the maximal order, which may have associated orders which are Hopf.

Chapter 8 gives two theorems of Byott: one, from {By95b}, giving conditions on when tame implies Galois, the other, from {By97c}, giving necessary congruence conditions on the break numbers, or ramification numbers, of a totally and wildly ramified Galois extension of local fields for the associated order to be Hopf. Our treatment is slightly different than that in the literature.

In {Gr92}, Greither classified (under suitable hypotheses) the R-Hopf orders in KG, where R is the valuation ring of a local field K containing \Bbb Q_p and G has order p^2. In Chapter 9 we give a direct construction of these Greither algebras, following {GC98}.

Chapter 10, from {Gr92} and {Ch95}, examines cyclic Galois extensions of local fields L/K of degree p^2 with suitable ramification numbers, and determines when the associated order is Hopf. The results show that the necessary congruence condition on ramification numbers for the associated order to be Hopf found in Chapter 8 is far from sufficient. Also in Chapter 10 is a classification of Greither orders which are realizable, that is, can be the associated order of the valuation ring of some cyclic Galois extension L/K of order p^2.

The results in Chapter 10, as also those in Chapter 6, depend on a description of the Hopf algebra orders over R. Thus much effort in recent years has gone into describing Hopf algebras over valuation rings of local fields. In chapter 11 we present one general strategy for constructing Hopf algebras, namely, via formal groups. The Oort Embedding Theorem states that any local abelian R-Hopf algebra arises from an isogeny of formal groups. Thus Chapter 11 introduces formal groups, reviews Lubin-Tate and other examples of formal groups, and discusses this result.

Chapter 12 then presents the Kummer theory of formal groups, from {CM94}, {Mo94}, {Mo96}, which shows that if a Hopf algebra H is constructed as in Chapter 11, then the principal homogeneous spaces for H, or equivalently, the Galois extensions for H^*, are easily and explicitly described. The results of chapters 8 and 12 are then combined to produce Byott's examples from {By99b}, cited above.

We have not included other results on constructing Hopf algebras over valuation rings: among recent works are results of Underwood {Un96} constructing Hopf orders in KC_{p^3}, and a memoir by the author with Greither, Moss, Sauerberg and Zimmermann {CGMSZ98}, and a thesis of H. Smith {Sm97}, constructing Hopf orders in group rings of elementary abelian p-groups.

It will be evident to a reader of this work that while much progress has been made in the past 15 years, there are many open questions which need to be answered before this subject is well-understood. Perhaps this exposition will help act as a resource and motivation for further work in the field.

Beyond a general background in algebra at the level of {Lg84}, the main prerequisite for this book is an acquaintance with some algebraic number theory. Serre's classic book, Local Fields (cited as {CL}), is a useful reference for local number theory.

This book focuses on results over local fields and their valuation rings. For a nice survey of some global results, including connections to L-functions, see {TB92}.

The author wishes to thank his students and collaborators over the years for their inspiration, motivation and mathematical insights, particularly Susan Hurley, Steve Tesser, David Weinraub, Rob Underwood, David Moss, Maureen Cox, Alan Koch, Tim Kohl, Hal Smith, ManYiu Tse, and Scott Carnahan, Cornelius Greither, Jim Sauerberg and Karl Zimmermann. Thanks to Ellen Fisher for her assistance in typing parts of the manuscript. Special thanks go to Nigel Byott for his careful reading of the manuscript, and for his numerous contributions to the subject: his influence on the mathematics presented in this book has already been noted, and will become clearer in subsequent chapters.

Albany, NY

February, 2000