LOCAL-GLOBAL PRINCIPLES FOR WITT RINGS

The paper investigates the connection between the Witt and Witt-Grothendieck rings of a field K and the corresponding rings of a given collection H of separable algebraic extensions of K. In particular, it deals with the question whether the former rings are the ''sheaf products'' of the latter rings. If H consists of pro-2 extensions and is closed in the natural topology then this happens precisely when the 2-Galois group of K is the free pro-2 product of the 2-Galois groups of the fields in H. This result is used to classify the quadratic forms over the field of totally real numbers, and more generally, over real-projective fields.