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Equations over Finite Fields: An Elementary Approach

Wolfgang M. Schmidt
Kendrick Press
Publication Date: 
Number of Pages: 
[Reviewed by
Donald L. Vestal
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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

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