Pseudo-discriminant and Dickson invarian

Abstract

1. Let E be a vector space of finite dimension over a field K. To a bilinear symmetric form f(x, y) defined over ExE is attached classically the notion of discriminant: it is an element of K which is not entirely defined by / ; however, it is entirely determined when in addition a basis of E is chosen, and when the basis is changed, the discriminant is multiplied by a square in K. More precisely, let u be a linear mapping of E into E, and let fx(x, y)=f(u(x), u(y)) the form transformed by ^ if Δ(f), Δ(f1) are the discriminants of / a n d f1 with respect to the same basis of E, and D(u) the determinant of u with respect to that basis, then one has the classical relation