Depending on your point of view, a function field might be the collection of meromorphic functions from a Riemann surface to the complex numbers, or it might be the set of regular functions on an algebraic curve, or it might be a finite extension of the field of rational functions over a fixed field. This last interpretation of function fields is possibly the richest as it gives an analogy between number fields and function fields which has led to a large amount of new mathematics in recent years. Gabriel Daniel Villa Salvador's book Topics in the Theory of Algebraic Function Fields was recently translated into English and published by Birkhäuser (the original version was published in Mexico in 2003). In the book, Salvador covers a large amount of material at a wide range of levels of difficulty, and as such his book could be used in a wide variety of settings.
Villa Salvador intended the first few chapters to be at a level that an undergraduate could follow. The topics in these chapters include algebraic and transcendental extensions, absolute values and valuations, divisors and class groups, and the Riemann-Roch theorem. The fourth chapter of the book is dedicated to examples and computations, and shows off the depth of the theory by looking at a variety of applications. The next few chapters would be quite appropriate for a graduate student or researcher in another field who wishes to learn about function fields. They begin with a primer on Galois theory and Dedekind domains, and then go on to discuss the special case of function fields defined over finite fields. This discussion includes an introduction to zeta-functions which leads into a whole chapter discussing the Riemann Hypothesis for Function Fields, including a full proof (which, for those of you unfamiliar with it, unfortunately does not work for the regular Riemann Hypothesis).
The later chapters for the book are probably best for advanced graduate students and researchers working in number theory. There is a chapter on the Riemann-Hurwitz formula which includes many examples, a chapter on cryptography which starts very basic but then introduces some technical hyperelliptic curve cryptosystems, several chapters on class field theory, and a chapter on Drinfeld modules, which give an explicit class field theory for function fields over finite fields and provide some steps along the road to the proof of the Langlands conjectures. The final chapter is about automorphism groups of function fields and the book also includes an appendix on group cohomology.
The entire book is very well written and Villa Salvador includes many examples to both illustrate and motive the theory being presented. There are also a handful of exercises in each chapter, although many of these merely direct the reader to "Prove Lemma 7.1.2" or the like, and they are probably the book's biggest weakness. The study of function fields is a beautiful area of mathematics which seems to be ever-increasing in importance, and this book is a wonderfully well-written introduction to the area.
Darren Glass is an assistant professor at Gettysburg College whose mathematical interests include number theory, Galois theory, and cryptography. He can be reached at email@example.com
Preface.- Algebraic and Numerical Antecedents.- Algebraic Function Fields of One Variable.- The Riemann–Roch Theorem.- Examples.- Extensions and Galois Theory.- Congruence Function Fields.- The Riemann Hypothesis.- Constant and Separable Extensions.- The Riemann–Hurwitz Formula.- Cryptography and Function Fields.- Introduction to Class Field Theory.- Cyclotomic Function Fields.- Drinfeld Modules.- Automorphisms and Galois Theory.- Appendix A: Cohomology of Groups.- Notations.- References.- Index.