Lectures on Finite Fields

Finite fields are fields with finitely many elements. The most familiar example is \(\mathbb{Z}/p\mathbb{Z}\), the integers modulo a prime \(p\), and general finite fields are always field extensions of \(\mathbb{Z}/p\mathbb{Z}\) and have a somewhat similar structure to the base field. Finite fields have applications in many areas of mathematics, including number theory, projective geometry, combinatorics, design of experiments, cryptography, and coding theory.

This book is a good introduction to the theory of finite fields, although it omits all the applications. Its approach is to go into a fair amount of depth on several different topics. There are excellent exercises at the end of each chapter that broaden the scope quite a bit. I think the omission of applications is the book’s biggest weakness.

The book is based on lecture notes from several different graduate courses, and is something of a “selected topics” book. It doesn’t intend to provide complete coverage There’s a strong number-theoretic flavor. There’s a short introductory chapter that deals with the nature and structure of finite fields. A plus for the book is that it does this without using Galois theory. There’s a whole chapter on algebraic number theory (number fields, not finite fields), that mostly deals with the structure of field extensions and with unique factorization; this material is needed in the rest of the book. There’s also a chapter on Gauss sums, that starts out in full generality dealing with characters of finite groups, then ends by dealing with the original Gauss sum of quadratic characters. Then about a third of the book deals with polynomials over finite fields and their factorization. It has a very nice exposition of Berlekamp’s factorization algorithm, a topic that is hard to find. It has some coverage of the Riemann Hypothesis for function fields. Then the book switches gears and spends the last chapter (also about a third of the book) on classical groups over finite fields. There’s a little bit of \(p\)-adic analysis spread throughout the book.

Most people would not need a whole book’s worth of information on finite fields, even a selective book like this one, so I think the audience is specialists. The prerequisites are not very strenuous; just a good understanding of abstract algebra. According to the back cover blurb, the book is intended as a text for a second year graduate course and as a reference for researchers. Usually when you study one of the application areas there is a brief (5 to 10 page) introduction to finite fields, and that’s all you need for the application. Some books with good brief coverage of this type include Ireland & Rosen’s A Classical Introduction to Modern Number Theory, von zur Gathen & Gerhard’s Modern Computer Algebra (lots of applications), and Dummit & Foote’s Abstract Algebra.

For those who really need extensive coverage, the “Bible” of finite fields is Lidl & Niederreiter’s Finite Fields (Cambridge, 2^nd edition 1997), which is a slightly revised version of the 1983 first edition from Addison–Wesley. This book is a monograph; there is also a “textbook” edition by the same authors titled Introduction to Finite Fields and Their Applications (Cambridge, 2^nd edition 1994). This “Bible” is thus about 35 years older than the book under review, and does not cover the most recent research. A comprehensive and more recent reference is Mullen & Panario’s 2013 Handbook of Finite Fields, which is definitely a reference and not a textbook, and assumes a great deal of prior knowledge.

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Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is allenstenger.com. His mathematical interests are number theory and classical analysis.