Real closures of fields at orderings of higher level (Q1122624)

The connections between orderings of higher level and the value groups of real-valued places are studied. The author gives valuation-theoretic characterizations of real closed fields and of real closures of fields at orderings of higher level. Let \(\tau\) : \(F\to {\mathbb{R}}\cup \{\infty \}\) be a real-valued place of the field F and let \(v:\quad F^.\to \Gamma\) be the valuation associated to the place \(\tau\). The author shows that there is a natural, one-to-one corresondence between the set of all orderings of F inducing the place \(\tau\) and the set of all subgroups \(\Delta\) of the value group \(\Gamma\) of v, such that \(\Gamma\) /\(\Delta\) is a finite cyclic group. Let \underbar{Iso} be the set of all F-isomorphism classes of real closures of F of higher exact level which induce on F the place \(\tau\). Let \b{P} denote the set of all preorders T of F containing \(\tau^{- 1}({\mathbb{R}}^{.2})\) and with v(T) of \(index\quad 2\) in \(\Gamma\) and let \b{F} denote the set of all filtrations \((\Gamma_ i)_{i\geq 0}\) of \(\Gamma\) by subgroups \(\Gamma_ i\) of \(\Gamma\) having \(\Gamma /\Gamma_ i\) cyclic of order \(2^ i\) for all \(i\geq 0\). The author shows that there is also a natural, one-to-one correspondence between \underbar{Iso} and \b{P}\(\times \underline F\). The set of real closures of F inducing a given ordinary ordering of F is also computed.