Abstract Algebra Theory and Applications - Computer Science ...

15.2 THE DIVISION ALGORITHM 263Corollary 15.5 Let F be a field. An element α ∈ F is a zero of p(x) ∈ F [x]if and only if x − α is a factor of p(x) in F [x].Proof. Suppose that α ∈ F and p(α) = 0. By the division algorithm, thereexist polynomials q(x) and r(x) such thatp(x) = (x − α)q(x) + r(x)and the degree of r(x) must be less than the degree of x − α. Since thedegree of r(x) is less than 1, r(x) = a for a ∈ F ; therefore,p(x) = (x − α)q(x) + a.But0 = p(α) = 0 · q(x) + a = a;consequently, p(x) = (x − α)q(x), and x − α is a factor of p(x).Conversely, suppose that x−α is a factor of p(x); say p(x) = (x−α)q(x).Then p(α) = 0 · q(x) = 0.□Corollary 15.6 Let F be a field. A nonzero polynomial p(x) of degree n inF [x] can have at most n distinct zeros in F .Proof. We will use induction on the degree of p(x). If deg p(x) = 0, thenp(x) is a constant polynomial and has no zeros. Let deg p(x) = 1. Thenp(x) = ax + b for some a and b in F . If α 1 and α 2 are zeros of p(x), thenaα 1 + b = aα 2 + b or α 1 = α 2 .Now assume that deg p(x) > 1. If p(x) does not have a zero in F , then weare done. On the other hand, if α is a zero of p(x), then p(x) = (x − α)q(x)for some q(x) ∈ F [x] by Corollary 15.5. The degree of q(x) is n − 1 byProposition 15.2. Let β be some other zero of p(x) that is distinct from α.Then p(β) = (β − α)q(β) = 0. Since α ≠ β and F is a field, q(β) = 0. Byour induction hypothesis, p(x) can have at most n − 1 zeros in F that aredistinct from α. Therefore, p(x) has at most n distinct zeros in F . □Let F be a field. A monic polynomial d(x) is a greatest commondivisor of polynomials p(x), q(x) ∈ F [x] if d(x) evenly divides both p(x)and q(x); and, if for any other polynomial d ′ (x) dividing both p(x) and q(x),d ′ (x) | d(x). We write d(x) = gcd(p(x), q(x)). Two polynomials p(x) andq(x) are relatively prime if gcd(p(x), q(x)) = 1.