Dividing polynomials using the resultant matrix

Abstract

Let f, g be polynomials over a Noetherian ring A. We use the matrix coming from the resultant of f and g to get a criterion for divisibility of f by g in terms of Fitting invariants as well as a method of dividing polynomials once we know g divides f. Further we show that this is equivalent to that cokernel or the image of the multiplication-by-g map on A[X]/(f) is free. As an application we show one can test irreducibility of an integral polynomial by computing minors of a matrix.