Topics in Galois Fields

There is general agreement that the study of finite fields began with the 1830 paper of Evariste Galois (Sur la theorie des nombres, Bulletin des Sciences Mathematiques XIII (1830), 428-435). It is for this reason that finite fields are also called Galois Fields in his honor.

 

Nowadays, finite fields have become an important area not only in their own right as an algebraic structure, but because of the role they play in modern applications.  We can find finite fields in finite geometries, combinatorics, and the analysis of algorithms, information and communication theory, signal processing, coding theory, cryptography, and computer science. They have concrete applications to databases, linear feedback shift registers, pseudo-random sequences, compact discs, DVDs and Blue-Ray discs, concert hall acoustics, radar and sonar, spread spectrum communication, antenna design, x-ray imaging, spectroscopy, symmetric and public-key cryptosystems, digital signatures, access control, and the design of telephone and computer networks.  Clearly, finite fields are an area of study that is rich in both theory and applications.

 

The book under review does what its title says: it covers topics in finite field theory.  It does this in a thorough, well written, and detailed way.  The book consists of 14 chapters that cover standard topics in finite fields along with some new results that have not yet appeared in textbooks.  There is moreover a large list of references dating from 1801 to 2019. The notation is clear and standard, which is a good aid to the reader.  The first chapter is a thorough review of topics in group theory and field theory.  The reader should have a background in Modern Algebra as found in Herstein’s Topics in Algebra or Jacobson’s three volumes on Modern Algebra, Linear Algebra and Field Theory.  As the authors state, the book is not intended as a textbook for an introductory course in field theory. Rather, it is a monograph on the single topic of finite fields.  

 

The topics covered include basic field theory, field extensions and the basics of Galois Theory, polynomial factorization, shift register sequences, characters and Gauss sums, to name a few.  There are also some new results included that have not yet appeared in textbooks.  These are the primitive normal basis theorem of Lenstra and Schoof, the theorem on the existence of primitive elements in affine hyperplanes by Cohen, and the Niederreiter method for factoring polynomials over finite fields.  The authors have included several algorithms for factoring polynomials over finite fields along with examples illustrating the algorithms.  These algorithms address the factoring methods of Berlekamp and Niederreiter. The authors go on to contrast these two methods. There are applications of finite fields and the Discrete Fourier Transform in Shift Register Sequences, characters of Discrete Fourier Transforms to difference sets.

 

There are a few exercises included in the text, although no solutions are provided.

 

The book will appeal to specialists in finite field theory and to readers with a knowledge of modern algebra who would like to learn about finite field theory. 

 

In summary, this text does what it claims to do. It is a well-crafted text on finite field theory and a welcome addition to existing finite field theory literature.

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Charles Traina is a professor of mathematics at St. John’s University in Jamaica, N.Y. His research interests are Group Theory and Measure Theory.