Polynomial Arithmetic

4.5. Polynomial Arithmetic

Before pursuing our discussion of finite fields, we need to introduce the interesting subject of polynomial arithmetic. We are concerned with polynomials in a single variable x, and we can distinguish three classes of polynomial arithmetic:

• Ordinary polynomial arithmetic, using the basic rules of algebra
• Polynomial arithmetic in which the arithmetic on the coefficients is performed modulo p; that is, the coefficients are in GF(p)
• Polynomial arithmetic in which the coefficients are in GF(p), and the polynomials are defined modulo a polynomial m(x) whose highest power is some integer n

This section examines the first two classes, and the next section covers the last class.

As an example, let f(x) = x³ + x² + 2 and g(x) = x² x + 1, where S is the set of integers. Then

f(x) + g(x) = x³ + 2x² x + 3

f(x) g(x) = x³ + x + 1

f(x) x g(x) = x⁵ + 3x² 2x + 2

Figures 4.3a through 4.3c show the manual calculations. We comment on division subsequently.