1 - Finite Fields and Coding Theory

Abstract

This chapter discusses the finite fields and coding theory. The problem of classifying the various types of extensions of a field and give the basic properties of such extensions are presented. An element is said to be algebraic of degree if it satisfies an irreducible polynomial of degree. The other zeros of are the conjugates of a, the terminology coming from the more usual concept of the roots of an irreducible quadratic over the reals. It is found that if every element is algebraic over K, then L is called an algebraic extension of K. It can be shown that any finite extension of a field is an algebraic extension. An extension that is not algebraic is called transcendental. An extension L of K is called simple if there exists an element a. If K has characteristic zero, any finite extension is both algebraic and simple.