Minimal forms wit respect to function fields of conics

Summary LetF be a field of characteristic ≠2, and let ϱ be an anisotropic conic overF. Anisotropic quadratic forms φ overF which become isotropic over the function fieldF(ϱ), but which do not contain proper subforms becoming isotropic, are calledF(ϱ)-minimal forms. It is investigated how upper bounds for the dimension ofF(ϱ)-minimal forms depend on certain properties and invariants of the fieldF. The existence of fieldsF and conics ϱ such thatF containsF(ϱ)-minimal forms of arbitrarily large (odd) dimension is proved. During the work on this article, the first author was a postdoc at the Institute for Experimental Mathematics, University of Essen, Germany, supported by a grant from the Deutsche Forschungsgemeinschaft This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag.