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

Submission: 2007, Apr 15

Let $F$ be a field of characteristic $\not= 2$. 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 $F$. We show that if $k$ is a field of characteristic $\not=2$ having at least two distinct quadratic extensions, 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.

2000 Mathematics Subject Classification: 13C12, 13F20

Keywords and Phrases: Quadratic form, Biquaternion algebra, Second residue map

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