Polynomials, rings and fields

In this chapter, we introduce two of the most common of all algebraic structures in mathematics, a ``ring'' and a ``field''.

These terms will be defined precisely in the next section so here we just say a few words of motivation. Roughly speaking, a ``ring'' is a set of elements having two operations, usually called addition and multiplication, which behaves in many ways like the integers. You can add and multiply elements in a ring but you can't (usually) divide them and get another element in the ring. Roughly speaking, a ``field'' is a set of elements having two operations, usually called addition and multiplication, which behaves in many ways like the rational numbers. You can add and multiply elements in a field and you can divide one element by any non-zero element to get another element in the field.

However, there exist rings and fields which, unlike the rationals, have only finitely many elements. Because they have only finitely many elements, they are of particular interest for applications to computer science and communication theory. However, their finiteness also forces them to have many properties which are not true for the integers or rational numbers.

Related to this is the study of polynomials. Polynomials not only help us to construct new and useful examples of rings and fields but are of interest in themselves. The ``ring'' of polynomials in one variable ^2.1has two binary operations - addition and multiplication. These operations have many properties similar to the integers and we can, and will, study analogs of many of the properties considered in chapter 1. In particular, we will study factorizations of polynomials, irreducible polynomials (which are the analogs of prime integers), and modular arithmetic of polynomials (the polynomial analog of addition and multiplication modulo an integer $m$). The modular arithmetic of polynomials leads to other rings of particular interest to coding theorists, studied further in the next chapter.

We let ${\mathbb{C}}$ denote the complex numbers, ${\mathbb{R}}$ denote the real numbers, and let ${\mathbb{Q}}$ denote the rational numbers. Let $\mathbb{Z}$ denote the integers and let $\mathbb{F}_p =\mathbb{Z}/p\mathbb{Z}$, if $p$ is a prime.