Abstract Algebra Theory and Applications - Computer Science ...

15.2 THE DIVISION ALGORITHM 26115.2 The Division AlgorithmRecall that the division algorithm for integers (Theorem 1.3) says that ifa and b are integers with b > 0, then there exist unique integers q and rsuch that a = bq + r, where 0 ≤ r < b. The algorithm by which q and rare found is just long division. A similar theorem exists for polynomials.The division algorithm for polynomials has several important consequences.Since its proof is very similar to the corresponding proof for integers, it isworthwhile to review Theorem 1.3 at this point.Theorem 15.4 (Division Algorithm) Let f(x) and g(x) be two nonzeropolynomials in F [x], where F is a field and g(x) is a nonconstant polynomial.Then there exist unique polynomials q(x), r(x) ∈ F [x] such thatf(x) = g(x)q(x) + r(x),where either deg r(x) < deg g(x) or r(x) is the zero polynomial.Proof. We will first consider the existence of q(x) and r(x). Let S ={f(x) − g(x)h(x) : h(x) ∈ F [x]} and assume thatg(x) = a 0 + a 1 x + · · · + a n x nis a polynomial of degree n. This set is nonempty since f(x) ∈ S. If f(x) isthe zero polynomial, then0 = f(x) = 0 · g(x) + 0;hence, both q and r must also be the zero polynomial.Now suppose that the zero polynomial is not in S. In this case thedegree of every polynomial in S is nonnegative. Choose a polynomial r(x)of smallest degree in S; hence, there must exist a q(x) ∈ F [x] such thatr(x) = f(x) − g(x)q(x),orf(x) = g(x)q(x) + r(x).We need to show that the degree of r(x) is less than the degree of g(x).Assume that deg g(x) ≤ deg r(x). Say r(x) = b 0 + b 1 x + · · · + b m x m and