Quadratic $D$-forms with applications to hermitian forms

Abstract

We study some properties of quadratic forms with values in a field whose underlying vector spaces are endowed with the structure of right vector spaces over a division ring extension of that field. Some generalized notions of isotropy, metabolicity and isometry are introduced and used to find a Witt decomposition for these forms. We then associate to every (skew) hermitian form over a division algebra with involution of the first kind a quadratic form defined on its underlying vector space. It is shown that this quadratic form, with its generalized notions of isotropy and isometry, can be used to determine the isotropy behaviour and the isometry class of (skew) hermitian forms.