Birational Composition of Quadratic Forms over a Local Field

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 $P$, $\operatorname{char}P\ne 2$.