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, char K ≠ 2. The problem of birational composition of f( X) and g( Y) is considered: When is the product f( X) · 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, char P ≠ 2. [ABSTRACT FROM AUTHOR]