Maximal ordered fields of rank n. II
[For part I of this paper see ibid. 17, 157-167 (1987; Zbl 0627.12016).] In this review by a field we mean an ordered field and by an extension we mean an ordered extension. The rank of a field F is defined to be the order type of the set of convex subgroups of the value group of F (and it is the same as the order type of the finest convex valuation ring of F). Given an extension K of a field F, there is a natural surjective map \(\psi_{K/F}\) from the set of convex valuation rings of K to that of F. An ordered field F is maximal if \(\psi_{K/F}\) is not bijective for any proper extension K of F. The main results of this paper are: \((1)\quad that\) for any field F there exists an extension K such that \(\psi_{K/F}\) is bijective and K is maximal ordered; and \((2)\quad that\) a maximal ordered field is isomorphic to the field of formal power series with real coefficients and exponents in a suitable totally ordered set.
scientific article; zbMATH DE number 4063226