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

Let $F$ be a field of characteristic $\neq 2$. 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 $\varphi\in I^3F$ 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 $\varphi\in I^2F$ 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.

In this article, we investigate how these properties $D(n)$ relate to each other and we show how one can construct fields which fail to have property $D(n)$, $n>2$, by starting with a field which fails to have property $D(2)$ and then passing to transcendental field extensions. Particular emphasis is devoted to the situation where $K$ is a field with a discrete valuation with residue field $k$ of characteristic $\neq 2$. Here, we study how the properties $D(n)$ behave when one passes from $K$ to $k$ or vice versa. We conclude with some applications and an explicit and detailed example involving rational function fields of transcendence degree at most four over the rationals.

1991 Mathematics Subject Classification: Primary11E04; Secondary 11E16, 11E81, 16K20.

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