Nicolas Grenier-Boley, Emmanuel Lequeu, Mohammad Mahmoudi: On hermitian Pfister forms,,

Submission: 2007, Sep 11

It is known that a quadratic Pfister form over a field 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 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.

2000 Mathematics Subject Classification: 11E39 (11E81)

Keywords and Phrases: isotropy, hyperbolicity, Pfister forms, multiquaternion algebras, u-invariant.

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