Characterizing Reduced Witt Rings of Fields

Let IV(F) denote the Mitt ring of nondegenerate symmetric bilinear forms over a field F. In this paper wc shall be concerned only with formally real fields, for which we write Wr,,l(F) ~mm W(F)/Wil W(F) for the reduced R’itt ring. In [13, 141 the rings W(F) and iTred are shown to be special cases of absfrart lWtt rirqs and a great deal of the ring structure is developed in this setting. In [6] it is shown that not all of these abstract Wtt rings can be FVitt rings of fields and more examples are given in [7]. In this paper we shall show precisel! which of the torsion fret abstract Witt rings (subject to a certain finiteness restriction) can be reduced Witt rings of fields. In Section 2 we give an inductive construction of all reduced Witt rings of fields with only finitely many places into the real numbers R. This construction provides a powerful tool for proving ring-theoretic facts about reduced \Vitt rings. We apply this construction in Section 3 to obtain an explicit description of the structure of these rings in terms of the real places on any field whose reduced Witt ring is isomorphic to the given ring. In Section 4 we look at another application of the indutcive construction. \$‘e pro\-e the following conjecture in the case that F is a field with only finitely many places into [w: If cp E W(F) ma s into PF, for each real closure p. F, of F, then q~ is in WJF) 1 IIF, w ere IF denotes the maximal ideal of all even h dimensional forms o\-cr F and W,(F) d enotes the torsion subgroup of W(F). Before we begin our inductive construction, we shall need some definitions and notation. As in [ 131, wc shall write X(F) or X( W,,,l(F)) for the Boolean space of orderings of a field F, and we shall think of Wred(F) as a subring of ‘&(X(F), Z), the ring of all continuous functions from -Y(F) to Z, where Z has the discrete topology. Recall that the topology of X(F) is induced by the Harrison subbasis, which consists of all sets of the form