Abstract Algebra Theory and Applications - Computer Science ...

15.1 POLYNOMIAL RINGS 25715.1 Polynomial RingsThroughout this chapter we shall assume that R is a commutative ring withidentity. Any expression of the formf(x) =n∑a i x i = a 0 + a 1 x + a 2 x 2 + · · · + a n x n ,i=0where a i ∈ R and a n ≠ 0, is called a polynomial over R with indeterminatex. The elements a 0 , a 1 , . . . , a n are called the coefficients of f.The coefficient a n is called the leading coefficient. A polynomial is calledmonic if the leading coefficient is 1. If n is the largest nonnegative numberfor which a n ≠ 0, we say that the degree of f is n and write deg f(x) = n.If no such n exists—that is, if f = 0 is the zero polynomial—then the degreeof f is defined to be −∞. We will denote the set of all polynomials withcoefficients in a ring R by R[x]. Two polynomials are equal exactly whentheir corresponding coefficients are equal; that is, if we letp(x) = a 0 + a 1 x + · · · + a n x nq(x) = b 0 + b 1 x + · · · + b m x m ,then p(x) = q(x) if and only if a i = b i for all i ≥ 0.To show that the set of all polynomials forms a ring, we must first defineaddition and multiplication. We define the sum of two polynomials asfollows. LetThen the sum of p(x) and q(x) isp(x) = a 0 + a 1 x + · · · + a n x nq(x) = b 0 + b 1 x + · · · + b m x m .p(x) + q(x) = c 0 + c 1 x + · · · + c k x k ,where c i = a i + b i for each i. We define the product of p(x) and q(x) to bewherec i =p(x)q(x) = c 0 + c 1 x + · · · + c m+n x m+n ,i∑a k b i−k = a 0 b i + a 1 b i−1 + · · · + a i−1 b 1 + a i b 0k=0for each i. Notice that in each case some of the coefficients may be zero.