On property $D(2)$ and common splitting field of two biquaternion algebras

A. S. Sivatski

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 14, Tome 338 (2006), pp. 242-250

Résumé

Let $F$ be a field of characteristic $\ne 2$. We say that $F$ has property $D(2)$ if for any quadratic extension $L/F$ and any two binary quadratic forms over $F$ having a common nonzero value over $L$ this value can be chosen in $F$. There exist examples of fields of characteristic 0 which do not satisfy property $D(2)$. However, as far as we know, such examples of positive characteristic have not been constructed. In this note we show that if $k$ is a field of characteristic $\ne 2$ such that $\|k^*/{k^*}^2\|\ge 4$, then for the field $k(x)$ property $D(2)$ does not hold. Using this we construct two biquaternion algebras over a field $K=k(x)((t))((u))$ such that their sum is a quaternion algebra, but they do not have a common biquadratic (i.e. a field of the kind $K(\sqrt a,\sqrt b)$, where $a,b\in K^*$) splitting field.