Book review

The theory of finite fields is very old but still very vigorous: there is a lot of research going on in our days. This is because finite fields have an important role in modern applications not only in other theoretical domains as combinatorics and finite geometries, but also in areas of information and communication theory, coding theory, cryptography, computer science and signal processing. The concrete applications are very spectacular and the familiarization with this theory had become a need for engineers and computer scientists.

The present volume is written by an active researcher and the content is based on lecture courses given by the author. The book is not intended to be a first introduction in the subject. Some basic facts are reviewed in the first chapter but the knowledge of the first two year general algebra is assumed and also introductory reading is required by the author.

The book is divided into 7 chapters, and let me enumerate them: 1. Basic results; 2. Explicit constructions of finite fields; 3. Normal bases; 4. Dual bases; 5. Normal bases and duality; 6. Shift register sequences; 7. Characters and Gaussian sums. The material deals with both theoretical and arithmetical aspects, and the main stress is put on computational problems (in applications one need sometimes to make efficient calculations in very large fields). Also a lot of examples illustrate the topics and the whole content is the best illustration of how abstract mathematics can be very concretely applied.

As a result, this volume is mainly recommended to graduate students in theoretical computer science and communication technology, and also to engineers who need a better understanding of the results which they apply.

Finally, the good index, the rich bibliography and the reasonable price should also be remarked. This volume will be an excellent investment for anyone who is interested in this fascinating subject.