Quadratic forms and Pfister neighbors in characteristic 2

Abstract:

We study Pfister neighbors and their characterization over fields of characteristic $2$, where we include the case of singular forms. We give a somewhat simplified proof of a theorem of Fitzgerald which provides a criterion for when a nonsingular quadratic form $q$ is similar to a Pfister form in terms of the hyperbolicity of this form over the function field of a form $\varphi$ which is dominated by $q$. From this, we derive an analogue in characteristic $2$ of a result by Knebusch saying that, in characteristic $\neq 2$, a form is a Pfister neighbor if its anisotropic part over its own function field is defined over the base field. Our result includes certain cases of singular forms, but we also give examples which show that Knebusch’s result generally fails in characteristic $2$ for singular forms. As an application, we characterize certain forms of height $1$ in the sense of Knebusch whose quasi-linear parts are of small dimension. We also develop some of the basics of a theory of totally singular quadratic forms. This is used to give a new interpretation of the notion of the height of a standard splitting tower as introduced by the second author in an earlier paper.