The reduced Witt ring of a formally real field

Abstract

Abstract.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.

Quadratic forms with prescribed Stiefel-Whitney invariants

“…Formula (1) above shows (iii) implies (i) (note that vv ((-1>) = 1 + /(-1)). That (i) implies (ii) follows from the fact that for any q E W(F) we have (2) w(q-(dimq)-l) = (l,w(q)). …”

“…We may suppose F is formally real (otherwise the lemma is trivially true) and n^l. (W τed (F) is 0-stable if and only if F has at most one ordering [1, 3.14], i.e., F has at most one place σ into R and |Aao.| = l [2].) Now suppose W τed (F) is n -stable.…”

“…If s(E)= 1, then £ admits at least two distinct places into R, say σ and r [2]. Then so again W red (F) is n-stable.…”

Characterizing reduced Witt rings. II

“…4* Applications. In [4], Brown attempted to characterize, for a formally real field F, the elements of ^(X, Z) which lie in the image of W τeά (F) using valuation theory. He was led to define the concept of an "exact" field.…”

The reduced Wittring

“…A The description of sign W(K) in C(X,Z') was also attacked by R. Brown [4] and settled for the case that K admits only finitely many real places. For the general case he was led to a conjecture which (in his terminology) states that all formally real fields are exact.…”