On 14-dimensional quadratic forms in $I^3$, 8-dimensional forms in $I^2$, and the common value property

Abstract

Let $F$ be a field of characteristic $\ne 2$. We define certain properties $D(n),n\in \{2,4,8,14\}$, of $F$ as follows : $F$ has property $D(14)$ if each quadratic form $\phi \in I^{3}F$ of dimension $14$ is similar to the difference of the pure parts of two 3-fold Pfister forms; $F$ has property $D(8)$ if each form $\phi \in I^{2}F$ of dimension 8 whose Clifford invariant can be represented by a biquaternion algebra is isometric to the orthogonal sum 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 always have (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, Wadsworth and the second author. In particular, fields where $D(2)$ does not hold have been known to exist.