Corollary 2

Tag 06FD, permanent. Cite it as [Bourbaki, Tag 06FD].

— For every integer $r \geq 0$ we have

\operatorname{res}_{p,q+r}(f, g) = t_p^r \cdot \operatorname{res}_{p,q}(f, g) .

Bearing in mind Example 1 (IV, p. 76) we need only take

q_1 = r, g_1 = 1

in (35).

Assume that f is monic, and let $\rho : A \to B$ be a ring homomorphism and $\xi_1, \ldots, \xi_p$ elements of B such that we have the decomposition

^\rho f(X) = (X - \xi_1) \ldots (X - \xi_p)

By Remark 2 (IV, p. 77) and Cor. 1 above we have

\rho(\operatorname{res}(f, g)) = g(\xi_1) \ldots g(\xi_p)

This remark applies in particular to the universal decomposition off (IV, p. 73) and since

\rho

is then injective, this provides a means of calculating

\operatorname{res}(f, g)

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.