On Hermitian Pfister Forms

Abstract

Let K be a field of characteristic different from 2. It is known that a quadratic Pfister form over K is hyperbolic once it is isotropic. It is also known that the dimension of an anisotropic quadratic form over K belonging to a given power of the fundamental ideal of the Witt ring of K is lower bounded. In this paper, weak analogues of these two statements are proved for hermitian forms over a multiquaternion algebra with involution. Consequences for Pfister involutions are also drawn. An invariant uα of K with respect to a non-zero pure quaternion of a quaternion division algebra over K is defined. Upper bounds for this invariant are provided. In particular an analogue is obtained of a result of Elman and Lam concerning the u-invariant of a field of level at most 2.

On split products of quaternion algebras with involution in characteristic two

“…If the algebra A is split, then there is a decomposition For the case where the characteristic of F is different from 2, many authors have studied this conjecture either in its current formulation or in its equivalent variations, including the Pfister factor conjecture in the theory of spaces of similarities (developed by Shapiro), Hurwitz problem of composition of quadratic forms and Pfister involutions (see [16], [18], [14], [1], [13], [15], [9], [8], [10]). Finally in [3], Becher proved this conjecture in the case of characteristic different from 2.…”