ABSTRACT
This chapter reviews some of the basic algebraic concepts that are probably familiar to many readers and introduces some topics for specific use in later chapters. Topics reviewed include permutation groups, the ring of integers, polynomial rings, and finite fields. The chapter also presents examples that incorporate these topics using the philosophies of concepts. There are many ways to express a permutation as a product of transpositions, and the number of transpositions in these expressions can vary. However, the number of transpositions in the expressions of a permutation as a product of transpositions must be always even or always odd. A permutation is said to be even if it can be expressed as the product of an even number of transpositions, and odd otherwise. The chapter also shows how MATLAB can be used to construct the nonzero elements as powers of x in a finite field for prime and primitive polynomial.
