Isotropy of multiples of Pfister forms and weak isotropy of forms over field extensions

Abstract

The isotropy of multiples of Pfister forms is studied. In particular, an improved lower bound on the value of their first Witt index is obtained. This result and certain of its corollaries are applied to the study of the weak isotropy index (or equivalently, the sublevel) of arbitrary quadratic forms. The relationship between this invariant and the level of the (quadratic) form is investigated. The problem of determining the set of values of the weak isotropy index of a form with respect to field extensions is addressed, with both admissible and inadmissible integers being determined. The analogous investigation with respect to the level of a form is also undertaken, with some questions asked by Berhuy, Grenier-Boley and Mahmoudi being resolved. An examination of the weak isotropy index and the level of round and Pfister forms concludes this article.