Proposition 8

— Suppose that f and g are monic of degrees p and q respectively. With the above notation, the A-module F is free with basis $(x^i y^j)_{0 \leq i < p,\ 0 \leq j < q}$ and we have

Put $E = A[X]/(f)$ and $E' = A[Y]/(g)$. By II, p. 253, Cor. 1, the homomorphism $\sigma$ of $E @ E'$ into F derived from the canonical homomorphism $A[X] \otimes A[Y] \to A[X, Y]$ is bijective; this proves the assertion about the basis of F. We now identify E with its image in F under $\sigma$. Then the E-algebra homomorphism of $E[Y]/(g(Y))$ into F which maps Y to y is an isomorphism.

By the transitivity of the norm (III, p. 546) we have

By Prop. 7 (IV, p. 77), $N_{F/E}(x - y)$ is the resultant of the polynomials g(Y) and x − Y in E[Y], hence equal to g(x) (IV, p. 78, Cor. 3). By formula (34) and Prop. 7 (IV, p. 77) we thus have