The Reduced Witt Ring of a Formally

The reduced Witt rings of certain formally real fields are computed here in terms of some basic arithmetic invariants of the fields. For some fields, including the rational function field in one variable over the rational numbers and the rational function field in two variables over the real numbers, this is done by computing the image of the total signature map on the Witt ring. For a wider class of fields, including all those with only finitely many square classes, it is done by computing the Witt rings of certain ultracompletions of the field and representing the reduced Witt ring as an appropriate subdirect product of the Witt rings of the ultracompletions. The reduced Witt rings of a still wider class of fields, including for example the fields of transcendence degree one and the rational function field in three variables over the real numbers, are computed similarly, except that the description of the subdirect product no longer involves only local conditions.