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Finite Fields, with Applications to Combinatorics

Kannan Soundararajan
Publication Date: 
Number of Pages: 
Student Mathematical Library
BLL Rating: 

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Allen Stenger
, on
This is a very well-done short textbook on finite fields, aimed at undergraduates without any abstract algebra background. The author has taught it to “highly motivated first-year students at Stanford who were potential honors math majors” (p. xi). It starts from the beginning and axiomatically develops groups, rings, polynomials, and fields, and then the theory of finite fields. The last third of the book is applications to combinatorics (mostly combinatorial number theory), and a complete explanation and proof of the AKS (AgrawalKayalSaxena) polynomial-time primality test.
You may wonder why we need a whole book on finite fields, when standard references (such as Dummit and Foote’s Abstract Algebra and Ireland and Rosen’s A Classical Introduction to Modern Number Theory) cover this subject in about ten pages. The reasons are (1) the present book starts from scratch and assumes no prior knowledge, and takes a leisurely pace; (2) most treatments have already developed Galois theory and treat finite fields as a special case; (3) the present book is rich in applications, where other treatments usually skimp this.
The writing is very clear, and there are abundant cross-references and a good index in case you want to start in the middle of things rather than reading straight through. In particular the book is valuable if you already know about finite fields but would like to see some interesting applications.  As abstract algebra texts go, this treatment is very concrete with lots of specific examples. The book has a strong number theory flavor and brings out how these abstract structures generalize the integers.
The exercises are excellent. There are many of them, and all are fairly challenging. They give further developments of ideas in the text or describe alternative approaches. Unfortunately, there are no hints or answers for the exercises.

Allen Stenger is a math hobbyist and retired software developer. He was Number Theory Editor of the Missouri Journal of Mathematical Sciences from 2010 through 2021. His personal website is His mathematical interests are number theory and classical analysis.