Remark

Tag 0DK9, permanent. Cite it as [Bourbaki, Tag 0DK9].

Algebra, chapter IX, § 2. Discriminant d'une forme sesquilinéaire

Suppose that B is an overfield of A. Let $S = (x_1, \ldots, x_n)$ be a basis of B, and $(s_1, \ldots, s_n)$ the A-isomorphisms of B into the algebraic closure $A'$ of A (each $s_j$ being repeated $[B : A]_i$ times). Recall that, for every $z \in B$ , one has $\mathrm{Tr}_{B/A}(z) = \sum_{j=1}^n s_j(z)$ (chap. VIII, § 12, No. 2, prop. 4).

It follows then from the formula for multiplication of determinants that one has

(7)\quad (\det(s_j(x_i)))^2 = D_{B/A}(x_1, \ldots, x_n).

This formula shows that Proposition 5 generalizes the separability condition given in chap. V, § 7, No. 2, Remark.

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

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