In 1976, Springer-Verlag published the first edition of this book, which is preserved here in Chapters 1 through 6. This includes the study of equations of various forms and systems of equations. As the title suggests, the approach here is elementary, although there is a small appearance of complex variables in the definition of the L-function for a function field. Most of the work involves abstract algebra: characters on finite groups, character sums, Gaussian sums, Kloosterman sums, absolute irreducibility, varieties, rational and birational maps. The ultimate goal in most of these situations is to provide a bound on the number of solutions a polynomial equation, or a system of polynomial equations, can have in a finite field. A large part of this section consists of the author’s proof of Weil’s results using an elementary approach. Some standard results, such as the Hasse-Davenport relation and the theorems of Chevalley and Warning, are proven here.
In this, the second edition, published by Kendrick Press, the author adds a second section, called Bombieri’s Version of Stepanov’s Method. The goal here is provide elementary proofs for Weil’s work using Bombieri’s approach. There are more topics from algebra and number theory: valuations and places, Hensel’s Lemma, the Riemann-Roch Theorem, and then a return to Zeta functions and ultimately, a proof of the Riemann hypothesis for function fields in one variable over a finite ground field.
The book is well documented, and could serve as a good resource for graduate students interested in equations over finite fields. A small warning: there is no index, so you’ll have to rely on the table of contents. But if you remember where things are, then you’ll have a decent reference book.
Donald L. Vestal is Associate Professor of Mathematics at Missouri Western State University. His interests include number theory, combinatorics, and a deep admiration for the crime-fighting efforts of the Aqua Teen Hunger Force. He can be reached at firstname.lastname@example.org