Abstract Algebra Theory and Applications - Computer Science ...

15PolynomialsMost people are fairly familiar with polynomials by the time they begin tostudy abstract algebra. When we examine polynomial expressions such asp(x) = x 3 − 3x + 2q(x) = 3x 2 − 6x + 5,we have a pretty good idea of what p(x) + q(x) and p(x)q(x) mean. We justadd and multiply polynomials as functions; that is,(p + q)(x) = p(x) + q(x)= (x 3 − 3x + 2) + (3x 2 − 6x + 5)= x 3 + 3x 2 − 9x + 7and(pq)(x) = p(x)q(x)= (x 3 − 3x + 2)(3x 2 − 6x + 5)= 3x 5 − 6x 4 − 4x 3 + 24x 2 − 27x + 10.It is probably no surprise that polynomials form a ring. In this chapter weshall emphasize the algebraic structure of polynomials by studying polynomialrings. We can prove many results for polynomial rings that are similarto the theorems we proved for the integers. Analogs of prime numbers, ofthe division algorithm, and of the Euclidean algorithm exist for polynomials.256