From the previous chapter, we know that a code alphabet A is a finite set. In order to play mathematical games, we are going to equip A with some algebraic structures. As we know, a field, such as the real field R or the complex field C, has two operations, namely addition and multiplication. Our idea is to define two operations for A so that A becomes a field. Of course, then A is a field with only finitely many elements, whilst R and C are fields with infinitely many elements. Fields with finitely many elements are quite different from those that we have learnt about before.

The theory of finite fields goes back to the seventeenth and eighteenth centuries, with eminent mathematicians such as Pierre de Fermat (1601–1665) and Leonhard Euler (1707–1783) contributing to the structure theory of special finite fields. The general theory of finite fields began with the work of Carl Friedrich Gauss (1777–1855) and Evariste Galois (1811–1832), but it only became of interest for applied mathematicians and engineers in recent decades because of its many applications to mathematics, computer science and communication theory. Nowadays, the theory of finite fields has become very rich. In this chapter, we only study a small portion of this theory. The reader already familiar with the elementary properties of finite fields may wish to proceed directly to the next chapter.