Isotropy of Products of Quadratic Forms over Field Extensions

The isotropy of products 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 form is investigated. The problem of determining the set of values of the weak isotropy index of a form as it ranges over field extensions is addressed, with both admissible and inadmissible integers being determined. An analogous investigation with respect to the level of a form is also undertaken. An examination of the weak isotropy index and the level of round and Pfister forms concludes this article.