Corollary 1

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

— (i) Let $\lambda, \alpha_1, \ldots, \alpha_p$ be elements of A and suppose that $f(X) = \lambda (X - \alpha_1) \ldots (X - \alpha_p)$ . We have

\operatorname{res}_{p,q}(f, g) = \lambda^q g(\alpha_1) \ldots g(\alpha_p) .

(ii) Let

\mu, \beta_1, \ldots, \beta_q

be elements of A and suppose further that

g(X) = \mu (X - \beta_1) \ldots (X - \beta_q)

. Then we have

\operatorname{res}_{p,q}(f, g) = \lambda^q \mu^p \prod_{\substack{1 \leq i \leq p \\ 1 \leq j \leq q}} (\alpha_i - \beta_j) .

Assertion (i) follows directly from the formulae (28), (32) and (36). Now Assertion (ii) follows from (i).

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

Cited by

Nothing.