Introduction to finite fields and their applications. rev. ed. (English) Zbl 0820.11072

Cambridge: Univ. Press,. xi, 416 p. (1994).
The standard reference for anybody working in finite fields still is the authors’ monumental “Finite fields” [Encycl. Math. Appl. 20 (Addison Wesley 1983; Zbl 0554.12010)]. In 1986, a textbook version appeared [Introduction to finite fields and their applications (Cambridge University Press, 1986; Zbl 0629.12016)] which differed from the original mainly by omitting the 160-page bibliography, the extensive (historical) notes for each chapter and some of the more theoretical parts (on exponential sums, equations and permutation polynomials); on the other hand, a considerably broader treatment of applications of finite fields was given. This “student” version proved very popular and useful as a textbook but has been out of print for some time now. We note that some other, more specialized texts on finite fields appeared in recent years; but all of these complement rather than replace the book under review. Therefore, the appearance of the revised edition is most welcome.
In particular, this reviewer is glad that his only major complaint about the first edition no longer applies: The bibliography has been expanded and (short) historical and bibliographical notes for each chapter have been added.


11Txx Finite fields and commutative rings (number-theoretic aspects)
11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
94A60 Cryptography
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
94B15 Cyclic codes
05B25 Combinatorial aspects of finite geometries