The model theory of ‘R-formal’ fields

Abstract

Let 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 $\text{W(K) →}\text{Z}$. In particular, a field L is formally real over an ordered field K if and only if there is a homomorphism $\text{ϕ}{}_1\text{: W(L)→}\text{Z}$ which extends the given ‘signature’ $\text{ϕ}{}_{\mathrm{K}}\text{: W(K)→}\text{Z}$. (E.g. $\text{ϕ}{}_{\mathrm{K}}\text{= ϕ}{}_1\text{, i}{}_{\mathrm{\ast }}\text{,}\text{where}\text{i}{}_{\mathrm{\ast }}\text{: W(K)1 → W(L)}$ 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 $\text{Z}$ 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 $\text{Z}$ is replaced by some finitely generated reduced Witt ring.