Twisted Pfister forms
Detlev W. Hoffmann
Let be a field of characteristic . In this paper we investigate quadratic forms over which are anisotropic and of dimension , such that in the Witt ring they can be written in the form where and are anisotropic - resp. -fold Pfister forms, . We call these forms twisted Pfister forms. Forms of this type with are of great importance in the study of so-called good forms of height 2, and such forms with also appear in Izhboldin's recent proof of the existence of -fold Pfister forms over suitable fields , for which the function field is not excellent over . We first derive some elementary properties and try to give alternative characterizations of twisted Pfister forms. We also compute the Witt kernel of a twisted Pfister form . Our main focus, however, will be the study of the following problems: For which forms does a twisted Pfister form become isotropic over ? Which forms are equivalent to (i.e., the function fields and are place-equivalent over ) ? We also investigate how such twisted Pfister forms behave over the function field of a Pfister form of the same dimension which then leads to a generalization of the result of Izhboldin mentioned above.
Cite this article
Detlev W. Hoffmann, Twisted Pfister forms. Doc. Math. 1 (1996), pp. 67–102DOI 10.4171/DM/3