Polynomial Algebra

In many digital signal processing applications, one is required to process data sequences defined over a given number system. This number system may either be a field or a ring of finite or infinite order. In this chapter, we present some fundamental results in the theory of polynomial algebra. The emphasis, of course, is on data sequences and the corresponding polynomials defined over finite fields and finite rings. We observe here that all the results valid for polynomials defined over finite fields are also valid for polynomials defined over infinite fields. The order of the field does not play a very critical role in the polynomial algebra. The important distinction to be drawn is between the polynomials defined over fields and rings. Many basic results in the polynomial algebra over a field are no longer valid for the polynomial algebra over a ring.