If you want to play football, you go to a football field. If you want to play cricket, you go to a cricket field. But what if you want to play with numbers (with all four operations of addition, subtraction, multiplication, and division)? You need a field of numbers. When numbers of elements in the field are finite, we get a finite field. The smallest field (known as the binary field) has only two elements: zero and one. Our computers are based on this smallest field. We can safely say that modern information and communication technology (ICT) is based on finite fields. It provides reliability in ICT using error correction techniques from coding theory and provides security using cryptographic techniques from cryptography. The soul of coding and cryptography is finite fields.

Are you ready to play with finite fields? If you want to get some basic knowledge from a book and then dive into finite fields, this book is for you. This is not a textbook, but it provides comprehensive and up-to-date information in all areas of finite fields. It is divided into three parts. Part 1 gives a basic introduction to finite fields in two chapters. It begins with a historical introduction of the subject in chapter 1, continues to tables of irreducible polynomials in chapter 2, and concludes with a summary of open-source software. Part 2 provides beautiful theoretical properties for playing with finite fields in 11 chapters, covering such topics as irreducible polynomials, primitive polynomials, bases, exponential and character sums, equations over finite fields, permutation polynomials, special functions, sequences over finite fields, and curves over finite fields. Part 3 discusses applications in combinatorics, algebraic coding theory, and cryptography in the first three chapters. The last chapter deals with new fascinating and emerging areas where we find applications of finite fields, including biology, quantum computing, and some new areas in coding theory (such as space-time codes and network coding).

This book is a must-read for every electrical engineer, computer scientist, mathematician, and even biologist. They will find a wealth of information and ideas that can be used to understand various systems, build beautiful things, and even enjoy the power of algebra!