Integer-valued quadratic forms and quadratic diophantine equations

We investigate several topics on a quadratic form Phi over an algebraic number field including the following three: (A) an equation xi Phi.(t)xi = Psi for another form Psi of a smaller size; (B) classification of + over the ring of algebraic integers; (C) ternary forms. In (A) we show that the "class" of such a xi determines a "class" in the orthogonal group of a form Theta such that Phi approximate to Psi circle plus Theta. Such was done in [S3] when Psi is a scalar. We will treat the case of nonscalar Psi, and prove a class number formula and a mass formula, both of new types. In [S5] we classified all genera of Z-valued Phi. We generalize this to the case of an arbitrary number field, which is topic (B). Topic (C) concerns some explicit forms of the formulas in (A) when Phi is of size 3 and Psi is a scalar.