The model theory of ‘R-formal’ fields

AbstractLet K be a field, and let W(K) denote its Witt ring of Quadratic Forms. It is well-known in the theory of Quadratic Forms that the orders of K correpond in a one to one way with all ring surjections W(K) → Z. In particular, a field L is formally real over an ordered field K if and only if there is a homomorphism ϕ1: W(L)→Z which extends the given ‘signature’ ϕK: W(K)→Z. (E.g. ϕK = ϕ1, i∗, where i∗: W(K)1 → W(L) is the functinal map.)Using the above, one may discuss the usual theory of formally real and real closed fields in terms of Witt rings, Knebusch in [6] has, in the above setting, given a remarkable new proof of the uniqueness of real closures. One might ask what happens when the Z above is replaced by some other ring R? That is the subject of this present note. In particular, we shall prove some algebraic and model theoretic analogues of well-known results for real closed fields, where the above Z is replaced by some finitely generated reduced Witt ring