Geodesic
Birational Composition of Quadratic Forms over a Local Field
Matematičeskie zametki, Tome 85 (2009) no. 5, pp. 661-670.

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Let $f(X)$ and $g(Y)$ be nondegenerate quadratic forms of dimensions $m$ and $n$, respectively, over $K$, $\operatorname{char} K\ne 2$. The problem of birational composition of $f(X)$ and $g(Y)$ is considered: When is the product $f(X)\cdot g(Y)$ birationally equivalent over $K$ to a quadratic form $h(Z)$ over $K$ of dimension $m+n$? The solution of the birational composition problem for anisotropic quadratic forms over $K$ in the case of $m=n=2$ is given. The main result of the paper is the complete solution of the birational composition problem for forms $f(X)$ and $g(Y)$ over a local field $P$, $\operatorname{char}P\ne 2$.
Keywords: quadratic form, anisotropic quadratic form, binary quadratic form, local field
Mots-clés : birational composition, birational composition, Hilbert symbol. 
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A. A. Bondarenko. Birational Composition of Quadratic Forms over a Local Field. Matematičeskie zametki, Tome 85 (2009) no. 5, pp. 661-670. http://geodesic.mathdoc.fr/item/MZM_2009_85_5_a2/

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