Counting generalized orders on not necessarily formally real fields (Q2566499)

In his earlier paper the author defined the notion of generalized order on a field which leads to the notion of extended absolute value [J. Algebra 169, No. 3, 751--774 (1994; Zbl 0826.12005)]. If \(\tau\) is a place on a field \(F\) associated with the valuation \(\nu\) and \(\overline{\varphi}\) is an absolute value on the residue class field \(\overline{F}\), then a \((\tau,\overline{\varphi})\)-order on \(F\) is a sequence \(G=(G(n))_{n>0}\) of subgroups of \(F^\bullet \) such that for all positive integers \(m\) and \(n\) we have \ 1) \(m| n\Rightarrow G(n)\subseteq G(m)\),\ 2) \(\tau^{-1}(1)\subseteq G(n)\),\ 3) \(\nu(G(n))=\nu F\),\ 4) \(\tau(\tau^{-1}(\overline{F}^\bullet )\cap G(n))\) is the topological closure of \(\overline{F}^{\bullet n}\) in \(\overline{F}^\bullet \) with the topology induced by \(\overline{\varphi}.\) The main result of this paper says that there is a pairing \(\Phi:X_{(\tau,\overline{\varphi})}\times X_{(\tau,\overline{\varphi})}\longrightarrow \text{Hom}(\nu F, \widehat{M})\), which is bijective in each variable, where \(X_{(\tau,\overline{\varphi})}\) denotes the set of all \((\tau, \overline{\varphi})\) orders on \(F\) and \(\widehat{M}\) is the \(\mathbb{Z}\)-adic completion of the multiplicative group of the completion of \(\overline{F}\) with respect to \(\overline{\varphi}.\) This result is a natural generalization of the well known fact that there exists a canonical pairing \(X\times X\longrightarrow \text{Hom}(\Gamma, \{1,-1\} )\) which is bijective in each variable for \(X\) being the set of orderings on a field \(F\) compatible with a given place \(\tau\) from \(F\) to \(\mathbb{R}\). In the last section of the paper the author counts the number of extensions of a given generalized order on \(F\) to a field extension \(K\) of \(F.\) The paper ends with an application of this result to finite degree extensions of formally \(p\)-adic fields of arbitrary \(p\)-rank.