Finite fields. 2nd ed.

*(English)*Zbl 0866.11069
Encyclopedia of Mathematics and Its Applications. 20. Cambridge: Cambridge University Press. xiv, 755 p. (1996).

The theory of finite fields finds its origins in the work of several eminent mathematicians of the 17th and 18th centuries with the general theory credited to Gauss and Galois. The emergence of discrete mathematics as an important mathematical discipline, both in theory and practice, makes this volume, the first devoted entirely to finite fields, an indispensable reference. Before embarking on a chapter by chapter outline, some important aspects and features of the book are noted. To quote from the editorial policy of the series:

Books in the Encyclopedia of Mathematics and Its Applications will cover their subject comprehensively. Less important results may be summarized as exercises at the ends of chapters. For technicalities, readers can be referred to the bibliography, which is expected to be comprehensive. As a result, volumes will be encyclopedic references or manageable guides to major subjects.

It is remarkable how well the volume meets this statement. Comprehensive in content, the volume nonetheless depends on only a first level course in linear algebra with the occasional need for some abstract algebra and analysis. Throughout the book, every opportunity has been taken to provide simple and elegant proofs to deeper and complex treatments of more general results found elsewhere. Numerous and well chosen examples are worked out in detail and interesting exercises, designed to either illustrate or extend material covered, are given. The extensive historical notes at the end of each chapter make both fascinating reading and an important contribution to placing material in context. The 160 page bibliography is an invaluable resource, providing paths to sources which might have otherwise gone unreferenced. A few brief comments on the contents of each chapter are given.

Chapter 1 on algebraic foundations summarizes the relevant properties of groups, rings, fields and field extensions that are required. The second chapter considers the structure of finite fields, including characterization of finite fields, roots of irreducible polynomials and the elementary properties of traces, norms and bases. Roots of unity are treated from the point of view of general field theory. Different ways of representing elements in the finite field are given and two proofs of Wederburn’s theorem shown. Chapters 3 and 4 give a comprehensive look at properties and constructions of irreducible polynomials, as well as factoring algorithms for polynomials. Criteria for the irreducibility of binomials and trinomials are given, with a section on the properties of linearized polynomials. Chapter 5 considers exponential sums, including Gauss, Jacobi and Kloosterman sums, giving elementary proofs of many deep results, restricted to polynomials. Equations over finite fields are treated in chapter 6, using the estimates for character sums developed in the previous chapter. Several questions on permutation polynomials are explored in Chapter 7, both univariate and multivariate. Chapter 8 on linear recurring sequences, perhaps the most comprehensive treatment of the subject available, includes a treatment of the Berlekamp-Massey algorithm as well as distribution properties of sequences. The applications of finite fields to be found in Chapter 9 are limited to brief treatments of linear and cyclic codes over finite fields, affine and projective planes, certain questions of combinatorics and linear modular systems. The final chapter gives several tables of irreducible polynomials and field representations.

This volume is an indispensable tool for the researcher in finite fields and their applications. It is a beautifully written and presented book, painstakingly compiled and thoroughly researched.

As far as could be determined, it is a direct reprinting of the 1983 volume printed by Addison-Wesley (see the review in Zbl 0554.12010) which has been unavailable for several years. Its absence has been an impediment to the further development of the area, now corrected with this most welcome reprinting.

Books in the Encyclopedia of Mathematics and Its Applications will cover their subject comprehensively. Less important results may be summarized as exercises at the ends of chapters. For technicalities, readers can be referred to the bibliography, which is expected to be comprehensive. As a result, volumes will be encyclopedic references or manageable guides to major subjects.

It is remarkable how well the volume meets this statement. Comprehensive in content, the volume nonetheless depends on only a first level course in linear algebra with the occasional need for some abstract algebra and analysis. Throughout the book, every opportunity has been taken to provide simple and elegant proofs to deeper and complex treatments of more general results found elsewhere. Numerous and well chosen examples are worked out in detail and interesting exercises, designed to either illustrate or extend material covered, are given. The extensive historical notes at the end of each chapter make both fascinating reading and an important contribution to placing material in context. The 160 page bibliography is an invaluable resource, providing paths to sources which might have otherwise gone unreferenced. A few brief comments on the contents of each chapter are given.

Chapter 1 on algebraic foundations summarizes the relevant properties of groups, rings, fields and field extensions that are required. The second chapter considers the structure of finite fields, including characterization of finite fields, roots of irreducible polynomials and the elementary properties of traces, norms and bases. Roots of unity are treated from the point of view of general field theory. Different ways of representing elements in the finite field are given and two proofs of Wederburn’s theorem shown. Chapters 3 and 4 give a comprehensive look at properties and constructions of irreducible polynomials, as well as factoring algorithms for polynomials. Criteria for the irreducibility of binomials and trinomials are given, with a section on the properties of linearized polynomials. Chapter 5 considers exponential sums, including Gauss, Jacobi and Kloosterman sums, giving elementary proofs of many deep results, restricted to polynomials. Equations over finite fields are treated in chapter 6, using the estimates for character sums developed in the previous chapter. Several questions on permutation polynomials are explored in Chapter 7, both univariate and multivariate. Chapter 8 on linear recurring sequences, perhaps the most comprehensive treatment of the subject available, includes a treatment of the Berlekamp-Massey algorithm as well as distribution properties of sequences. The applications of finite fields to be found in Chapter 9 are limited to brief treatments of linear and cyclic codes over finite fields, affine and projective planes, certain questions of combinatorics and linear modular systems. The final chapter gives several tables of irreducible polynomials and field representations.

This volume is an indispensable tool for the researcher in finite fields and their applications. It is a beautifully written and presented book, painstakingly compiled and thoroughly researched.

As far as could be determined, it is a direct reprinting of the 1983 volume printed by Addison-Wesley (see the review in Zbl 0554.12010) which has been unavailable for several years. Its absence has been an impediment to the further development of the area, now corrected with this most welcome reprinting.

Reviewer: Ian F. Blake (Palo Alto)

##### MSC:

11Txx | Finite fields and commutative rings (number-theoretic aspects) |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11T06 | Polynomials over finite fields |

11T30 | Structure theory for finite fields and commutative rings (number-theoretic aspects) |

11T23 | Exponential sums |

11T24 | Other character sums and Gauss sums |

11T55 | Arithmetic theory of polynomial rings over finite fields |

11Y16 | Number-theoretic algorithms; complexity |

12E20 | Finite fields (field-theoretic aspects) |

51E15 | Finite affine and projective planes (geometric aspects) |

94B05 | Linear codes, general |

94B15 | Cyclic codes |

11T71 | Algebraic coding theory; cryptography (number-theoretic aspects) |

94A24 | Coding theorems (Shannon theory) |