Orders and order closure for not necessarily formally real fields (Q1340246)

An extended absolute value of a field \(F\) is a function \(\varphi: F\to \mathbb{R}\cup \{\infty \}\) which is a composition of a place \(\tau: F\to \overline {F}\cup \{\infty \}\) on \(F\) with an absolute value on the residue class field \(\overline {F}\) of the place. Let \(\nu\) denote the valuation induced by the valuation ring \(\varphi^{-1} (\mathbb{R})\). A \(\varphi\)-order on \(F\) is a sequence \(G= (G (n))_{n>0}\) of subgroups of \(F^\times\) such that for all positive integers \(m\) and \(n\), (1) if \(n\) divides \(m\), then \(G(m) \subset G(n)\), (2) \(1+ \varphi^{-1} (0) \subset G(n)\), (3) \(\nu (G(n))= \nu F\), (4) \(\tau (G(n)) \cap \overline {F}^.\) is the topological closure of \(\overline {F}^{\times n}\) in \(\overline {F}^\times\). The author uses extended absolute values and \(\varphi\)- orders to present a general setting for a fully unified treatment of the theories of formally real and formally \(p\)-adic fields. A closure of \(F\) with respect to an extended absolute value \(\varphi\) (\(\varphi\)-closure) is a maximal algebraic extension \(K\) of \(F\) with extended absolute value \(\psi\) such that \(\psi|_F= \varphi\) and the residue class field \(\overline {F}\) is dense in the residue class field \(\overline {K}\). The main theorem of the paper states: The correspondence \(K\mapsto (K^n \cap F^\times)_{n>0}\) induces a bijection from the set of \(F\)- isomorphism classes of \(\varphi\)-closures of \(F\) to the set of \(\varphi\)- orders of \(F\). This result generalizes the well-known theorem of Artin and Schreier on the bijective correspondence between orderings and isomorphism classes of real closures and implies the analogue of this theorem for formally \(p\)-adic fields which was not proved before. Considering Henselian extensions of fields with extended absolute value, the author generalizes some results of \textit{E. Becker} [Rocky Mt. J. Math. 14, 881-897 (1984; Zbl 0563.12023)] and of \textit{A. Prestel} and \textit{P. Roquette} [Formally \(p\)-adic fields, Lect. Notes Math. 1050 (Springer 1984; Zbl 0523.12016)] concerned with real closed fields and Henselian \(p\)-adic fields.