ABSTRACT
This chapter studies finite fields by determining the possible sizes or cardinalities of a finite field and illustrates how to construct a finite field. A primitive polynomial is a polynomial each of whose roots is a primitive element in the field. The chapter discusses several important properties of finite fields. The trace function is of fundamental importance in the study of finite field theory. It is also useful in various applications of finite fields. The chapter describes various properties related to polynomials over finite fields and shows that every function defined from a finite field to itself can be represented by a polynomial with coefficients in that finite field. It also discusses permutation polynomials, a class of polynomials that have various applications in combinatorics and cryptography. The chapter shows how to use finite fields and polynomials over finite fields to construct sets of mutually orthogonal Latin squares.
