1 - Finite Fields and Coding Theory

Abstract

Many introductory books on algebra contain a section on finite fields and prove some of their basic properties. Often their interest is to discuss Wedderburn's theorem on finite division rings or to consider the structure of groups of transformations of vector spaces over fields. For these ends, only the elementary properties of finite fields are needed. This chapter presents a more detailed account of the theory of finite fields, including material on polynomials over finite fields and linear transformations of vector spaces over finite fields. An introductory course of algebra allows an efficient development of the properties of finite fields and places them in their proper algebraic perspective as particular and interesting examples of a more general theory. The chapter reviews a few of the theories of fields, extensions of fields, and polynomials over fields. The theory of finite field is used in the construction and analysis of codes.