Detlev W. Hoffmann and Jean-Pierre Tignol

摘要

Let $F$ be a field of characteristic $eq 2$. We define certain properties$D(n)$, $nin{ 2,4,8,14}$, of $F$ as follows,: $F$ hasproperty $D(14)$ if each quadratic form $arphiin I^3F$ of dimension$14$ is similar to the difference of the pure parts of two $3$-foldPfister forms; $F$ has property $D(8)$ if each form $arphiin I^2F$ of dimension $8$ whose Clifford invariant can berepresented by a biquaternion algebra is isometric to the orthogonalsum of two forms similar to $2$-fold Pfister forms; $F$ has property $D(4)$ if any two $4$-dimensional forms over $F$ of the same determinant which become isometric over some quadratic extension alwayshave (up to similarity) a common binary subform; $F$ has property $D(2)$ if for any two binary forms over $F$ and for any quadratic extension$E/F$ we have that if the two binary forms represent over $E$ a common nonzero element, then they represent over $E$ a common nonzero element in $F$.Property $D(2)$ has been studied earlier by Leep, Shapiro, Wadsworthand the second author. In particular, fields where $D(2)$ does not hold have been known to exist.