Existence of SAP extension fields

Let F be a field. We shall let W(F) denote the Wit t ring of equivalence classes of nondegenerate symmetric bilinear forms over F. There are many questions one migh t ask about the extent to which one can prescribe the Wit t ring of an extension field of F and the natural homomorphism between these Wit t rings induced by the field inclusion. In this note we consider a formally real field F and its topological space of orderings X(F). There is a canonical homomorphism of W(F) into ~ (X(_F), Z), the ring of continuous functions from X(_F) into the integers, with kernel Wt(F), the torsion subgroup of W(_F) [8]. The image of this map (which is isomorphic to the reduced Wit t ring of F) is, in fact, contained in Z + ~(X(.F), 2Z). I f the image of the map is all of Z -b ~ (X(F) , 2E), we say tha t F satisfies the strong approximation proper ty (SAP). This is equivalent to saying that, given any clopen (closed and open) subset U of X(F), there is an element of F which is positive with respect to each ordering in U and negative with respect to each ordering not in U ([8, Cot. 3.21], [6]). We shall show tha t for an arbi trary formally real field F, one can find an extension field K satisfying SAP to which each ordering extends uniquely and such tha t the map W(_F) -->Z AW(X(F), 2E) is canonically the same as the homomorphism W(.F) ---> W(K) induced by the inclusion of F in K. We shall also see tha t K cannot always be chosen to be an algebraic extension of F.