Function fields of Pfister neighbours
A Pfister neighbor is a quadratic form \(Q\) of dimension \(m\) similar to a subform of a \(2n\)-dimensional Pfister form with \(n<m\). The authors introduce special Pfister neighbors. These are forms \(Q\) similar to forms of the shape \(P_0\perp aP_1\), where \(P_0\) is a Pfister form and \(P_1\) is a nonzero subform of \(P_0\). The Pfister form \(P=P_0\perp aP_0\) is uniquely determined by \(Q\) and is said to be the associated Pfister form of \(Q\) (and \(Q\) is a special Pfister neighbor of \(P\)). It is proved here that for an anisotropic Pfister form \(P\) any two special Pfister neighbors of \(P\) with equal dimensions have the same function field. (It is unknown whether this statement remains true after dropping the assumption that the neighbors are special.) An example is given of an 8-dimensional Pfister form with two 5-dimensional anisotropic special Pfister neighbors which are not similar. This produces an example of 5-dimensional anisotropic forms which have the same function field but are not similar. On the other hand, \textit{A. R. Wadsworth} [Trans. Am. Math. Soc. 208, 352-358 (1975; Zbl 0336.15013)]\ proved that two forms of dimension \(\leq 4\) which have the same function field are similar. Another result of the authors state that all Pfister neighbors of codimension \(\leq 4\) of an anisotropic Pfister form are special. An example is given of a codimension 5 Pfister neighbor (of a 16-dimensional Pfister form) which is not special.
scientific article; zbMATH DE number 840723