Cuts of ordered fields (Q1095973)

The rank of an ordered field F is defined to be the rank of the smallest convex valuation ring \(A(F,Q)=\{a\) in F such that \(| a| <b\) for some b in \(Q\}\). Given two ordered fields (F,\(\sigma)\) and (K,\(\tau)\) with \(F\subset K\), we say that K is an extension of F, if K is an extension field of F and the restriction of the ordering \(\tau\) to F is \(\sigma\). Suppose (F(x),\(\tau)\) is an extension of (F,\(\sigma)\) where x is a transcendental over F, we define a cut \(g(\tau)=(C,D)\) where \(C=\{a\) in F with \(a<x\}\) and \(D=\{a\) in F with \(a>x\}\). In this paper, for a real closed field F of finite rank n, the author defines subsets \(W_ i\), \(1\leq i\leq n+1\), of the set of all cuts of F, and proves that for any ordering \(\tau\) of F(x), g(\(\tau)\) is in \(W_ i\), is equivalent to the existence of distinct convex valuation rings B and B' of F(x) with respect to \(\tau\) such that \(A_ i=B\cap F=B'F\) where the \(A_ i's\) form an ascending chain of compatible valuation rings of F. He also shows that for a real closed field F, F is a maximal ordered field of rank n if and only if the set of all cuts of f is the union of all these subsets \(W_ i\).