Applications of Systems of Quadratic Forms to Generalised Quadratic Forms

Abstract

A system of quadratic forms is associated to every generalised quadratic form over a division algebra with involution of the first kind in characteristic two. It is shown that this system determines the isotropy behaviour and the isometry class of generalised quadratic forms. An application of this construction to the Witt index of generalised quadratic forms is also given.

Quadratic descent of generalized quadratic forms

“…Throughout this section, (D, θ) is a finite dimensional division algebra with involution of the first kind over a field F . We recall some constructions forms [9].…”

“…Also, as in [9], we say that Q is strongly nonsingular if q i is nonsingular for some i, and totally nonsingular if every q i is nonsingular.…”

“…We now fix (D, θ) as a finite dimensional division algebra with involution of the first kind over a field F . As in [9]…”

“…Fixing an F -basis B of D and letting B * ⊆ Hom F (D, F ) denote its dual basis, a system of quadratic forms q h := {ϕ • h | ϕ ∈ B * } was associated to h in [6] and used to study the isometry class and the isotropy behaviour of hermitian forms. Similarly, for a generalized quadratic space (V, ρ) over (D, θ), a system of quadratic forms q ρ was defined in [9].…”