Finite Fields and Polynomials

Abstract

Finite fields give rise to particularly useful and, in our view, beautiful examples of the applicability of rings and fields. Such applications arise both within mathematics and in other areas; for example, in communication theory, in computing and in statistics. In this chapter we present the basic properties of finite fields, with special emphasis on polynomials over these fields. The simplest finite field is the field \( {\mathbb{F}_2}\)consisting of 0 and 1, with binary addition and multiplication as operations. Many of the results for \( {\mathbb{F}_2}\) can be extended to more general finite fields.