Corollary 7

Tag 076E, permanent. Cite it as [Bourbaki, Tag 076E].

We have $\det(\operatorname{Tr}(u^{i+j})_{0 \leq i,j \leq n}) = \operatorname{dis}(\chi_u)$ .

Let D be the Vandermonde matrix $(\alpha_i^j)_{1 \leq i,j \leq n}$ . Then (III, p. 532, formula (29)):

\det(D)^2 = \prod_{i < j} (a_j - \alpha_i)^2 = \operatorname{dis}(\chi_u).

Moreover, the $(i, j)$ -th entry of $D \cdot 'D$ is $\sum_k \alpha_k^{i+j-2} = \operatorname{Tr}(u^{i+j-2})$ , and the corollary follows.

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

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