1. Theory of Non-Commutative Polynomials
2. Outlined by Cayley in P. G. Tait's Quaternions, third edition, Cambridge Press, 1890, pp. 157–159.
3. Cf., Bôcher's Introduction to Higher Algebra, Macmillan, 1927, problem 4 on p. 239, and the theorem on p. 202. The theorem on p. 202 is not precisely what we want here, since we need a proposition about polynomials in two variables. Two polynomials f(x, y), g(x, y) have two resultants, Rx obtained by eliminating x, and Ry by eliminating y. Bôcher's proof shows that the vanishing of Rz identically is a necessary and sufficient condition for the two polynomials to have a common factor involving x. But it is possible for f(x, y) and g(x, y) to have a common factor in y alone, with Rx not identically zero; in such a case R vanishes (e.g., consider the polynomials xy-4x-3y+12, xy+y2-4x-4y).