The most powerful and important ideas of coding theory are based on the arithmetic systems of Galois fields. Since these arithmetic systems may be unfamiliar to many, we must develop a background in this branch of mathematics before we can proceed with the study of coding theory.

In this chapter, we return to the development of the structure of Galois fields begun in Chapter 2. There we introduced the definition of a field, but did not develop procedures for actually constructing Galois fields in terms of their addition and multiplication tables. In this chapter, we shall develop such procedures. Galois fields will be studied by means of two constructions: one based on the integer ring and one based on polynomial rings. Because the integer ring and the polynomial rings have many properties in common we will find that the two constructions are very similar. Later, after the constructions of Galois fields are studied, we shall prove that all finite fields can be constructed in this way.

The integer ring

The set of integers (positive, negative, and zero) forms a ring under the usual operations of addition and multiplication. This ring is conventionally denoted by the label Z. We shall study the structure of the integer ring in this section.