TABLE OF CONTENTS In this chapter we introduce the concept of the resultant of two polynomials. The resultant has applications in calculations with algebraic numbers, the factorization of polynomials with coefficients in algebraic number fields, the solution of systems of polynomial equations, and the integration of rational functions.
The resultant of polynomials u and v is formally defined in Section 7.1 as the determinant of a matrix whose entries depend on the coefficients of the polynomials. In addition, we describe a Euclidean type algorithm that obtains the resultant without a determinant calculation. In Section 7.2, we use resultants to find polynomial relations for explicit algebraic numbers.
Suppose that we are given two polynomials
and want to know for which values of t the polynomials have a common factor with positive degree. Since the polynomials do not have a common factor for all values of t, a greatest common divisor algorithm cannot be used for this purpose. In this section, we introduce the resultant of two polynomials which provides a way to determine t.
Let s generalize things a bit. Let u and v be polynomials in F[x] with degree 2:
Our goal is to find a function of the coefficients of the two polynomials that determines if there is a common factor. Suppose, for the moment, that gcd (u, v)=1. Then, by Theorem 4.36 on page 137, there are unique polynomials A(x) and B(x) with deg(A)
