Browse Books

Reviewer: George James Davis

This is a textbook edition of the monograph Finite Fields [1]. The emphasis is on applications, and thus many of the more theoretical topics found in that earlier volume are not included here. The stated prerequisites include a first course in linear algebra, and some basic ideas in analysis. A background in algebra would, of course, be helpful, but is not essential. A brisk course in abstract algebra is given in Chapter 1, along with a rich set of exercises to solidify these important ideas. Students meeting only the minimal prerequisites may find the pace to be difficult, but those with stronger backgrounds could review this material rapidly. Chapter 2 discusses the structure of a finite field, including theorems on representation, extension, irreducible polynomials, and roots of unity. Wedderburn's Theorem is included as an optional section. One of the best features of this book is its problem sets. Students simply must understand these first two chapters to make any headway in the text; working the exercises should aid the task greatly. The next two chapters discuss polynomials over finite fields. Chapter 3 studies the notions of order, primitive polynomials, construction of irreducible polynomials, linearized polynomials and related topics. Chapter 4 covers concepts related to polynomial factorization. Algorithms are given to factor a polynomial in a “small” and a “large” finite field, and to find roots of polynomials. More good problem sets follow. The relatively short Chapter 5 explores some of the theory of exponential sums and, in particular, Gaussian sums. This is one of the chapters that was shortened from the Encyclopedia volume. Sequences where the terms depend linearly on a fixed number of predecessors are covered in detail in Chapter 6. Implementation on feedback shift registers, impulse response sequences, periodicity, characterization, and distribution properties are all given treatment. Another very good set of problems is found at the end. The last three chapters are devoted to applications. Chapter 7 covers some of the applications of finite fields to other areas of mathematics, notably affine and projective geometry, combinatorics, linear modular systems, and simulation of randomness. Applications to coding theory are discussed in Chapter 8, including cyclic codes, Bose-Ray-Chaudhuri-Hocquenghem codes, and Goppa codes. Several cryptosystems and a short background of cryptology are presented in Chapter 9. A comprehensive set of (often challenging) problems is found at each chapter's end. This is a very well-written book. The authors do a good job of telling the reader where we have been and where we are going. The application chapters are particularly well done. There is much material in this text, certainly more than can be covered in one quarter or semester. The interdependent structure of the chapters gives instructors considerable freedom in fitting the course to the students' needs. Anyone interested in a good textbook covering these topics should certainly review this book as a possible candidate.