This book is designed as a textbook edition of our monograph Finite Fields which appeared in 1983 as Volume 20 of the Encyclopedia of Mathematics and Its Applications. Several changes have been made in order to tailor the book to the needs of the student. The historical and bibliographical notes at the end of each chapter and the long bibliography have been omitted as they are mainly of interest to researchers. The reader who desires this type of information may consult the original edition. There are also changes in the text proper, with the present book having an even stronger emphasis on applications. The increasingly important role of finite fields in cryptology is reflected by a new chapter on this topic. There is now a separate chapter on algebraic coding theory containing material from the original edition together with a new section on Goppa codes. New material on pseudorandom sequences has also been added. On the other hand, topics in the original edition that are mainly of theoretical interest have been omitted. Thus, a large part of the material on exponential sums and the chapters on equations over finite fields and on permutation polynomials cannot be found in the present volume.

The theory of finite fields is a branch of modern algebra that has come to the fore in the last 50 years because of its diverse applications in combinatorics, coding theory, cryptology, and the mathematical study of switching circuits, among others.