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**Finite fields. Foreword by P. M. Cohn.**
*(English)*
Zbl 0554.12010

Encyclopedia of Mathematics and Its Applications, Vol. 20. Reading, Massachusetts etc.: Addison-Wesley Publishing Company. Advanced Book Program (1983); Cambridge etc.: Cambridge University Press. xx, 755 p. (1984).

The origins of the theory of finite fields and its connections with number theory reach back into the 17th and 18th century. In recent decades this subject has been rapidly growing in importance because of its diverse applications in such areas as coding theory, combinatorics, and the mathematical study of switching circuits. This book presents both the classical and the application-oriented aspect of the theory of finite fields.

Chapters 1 and 2 cover the background and the general structure theory of finite fields. Chapters 3 and 4 are devoted to polynomials and factorization algorithms. Chapters 5, 6, and 7 deal with exponential sums, equations, and permutation polynomials. Chapters 8 and 9 are devoted to linear recurring sequence and other applications.

The notes at the end of each chapter are excellent historical surveys. The bibliography is very comprehensive and up to date. This volume can be strongly recommended as the handbook of finite fields. In order to make the book useful as a text for advanced level courses the authors have inserted worked-out examples at appropriate points. These and the exercises at the end of each chapter are well-chosen.

Chapters 1 and 2 cover the background and the general structure theory of finite fields. Chapters 3 and 4 are devoted to polynomials and factorization algorithms. Chapters 5, 6, and 7 deal with exponential sums, equations, and permutation polynomials. Chapters 8 and 9 are devoted to linear recurring sequence and other applications.

The notes at the end of each chapter are excellent historical surveys. The bibliography is very comprehensive and up to date. This volume can be strongly recommended as the handbook of finite fields. In order to make the book useful as a text for advanced level courses the authors have inserted worked-out examples at appropriate points. These and the exercises at the end of each chapter are well-chosen.

Reviewer: Aimo Tietäväinen (Turku)

### 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 theory) |

94B15 | Cyclic codes |

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

94A24 | Coding theorems (Shannon theory) |