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Review: Finite Fields for Computer Scientists and Engineers June 17, 2011

Posted by flashbuzzer in Research.
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I recently finished “Finite Fields for Computer Scientists and Engineers” by R.J. McEliece. I originally used this book for a course taught by Dr. McEliece in the Winter 2002-03 quarter.

This book has two reviews on Amazon, so I hope to provide some additional insights for people who are considering obtaining the book.

In this book, the author presents the theory of finite fields with an eye towards their application in the design of communication systems. He begins by introducing the concept of a Euclidean domain along with the familiar – yet vital – construct of Euclid’s algorithm, which he then uses to show how primes can be uniquely factored in a Euclidean domain. With these results in hand, he shows how Euclidean domains can be used to construct fields, which forms a nice transition to the book’s core material: a discussion of the properties of finite fields, including existence and uniqueness, and how these properties can be used to factor polynomials over these fields. The rest of the book forms a set of advanced topics that draw on these fundamental concepts, including

  • the notion of the trace (relative to a subfield)
  • properties of linear recurrences over finite fields
  • the fundamentals of m-sequences and their cross-correlation properties.

The author skillfully motivates and presents the central results of this book. Indeed, his self-deprecating writing style triggered many fond memories from my days as an undergrad, as I took three courses from him. Somehow the author pulls off the difficult balancing act of presenting mathematically challenging material without truly sacrificing mathematical rigor, which is quite remarkable. The book also contains several fascinating concepts that I enjoyed learning, including Euler’s phi function, the Mobius inversion formula and the Chevalley-Warning theorem.

One aspect of the book that I did find to be a bit bothersome, though, was that each chapter was not divided into separate sections. In my opinion, this would have improved the flow of the book and provided many helpful “signposts” for the reader, especially while studying a rather lengthy chapter. More often than not I would be shown a result, eventually understand the proof, and then be unsure as to its importance in the context of the chapter in question.

Overall I would strongly recommend this book to those who want a gentle introduction to finite fields while gaining a healthy appreciation for their theoretical underpinnings.