Definition 1

Given a sesquilinear form $\Phi$ for $J$ on $E$ and a system $S = (x_1, \ldots, x_n)$ of $n$ elements of $E$, the discriminant of $\Phi$ with respect to this system is called, and is denoted by $D_\Phi(x_1, \ldots, x_n)$ or $D_\Phi(S)$, the element $\det(\Phi(x_i, x_j))$ of $A$.

If $(e_1, \ldots, e_n)$ is a basis of $E$, the discriminant of $\Phi$ with respect to this basis is none other than the determinant of the matrix of $\Phi$ with respect to this basis.

It follows from the definition of the extension of $\Phi$ to $\wedge^n E$ ($§ 1$, no 9) that we have

where $\Phi_{(n)}$ denotes the extension of $\Phi$ to $\wedge^n E$. For every permutation $\sigma \in \mathfrak{S}_n$, we therefore have