An Isomorphism Theorem for Real-Closed Fields

Abstract

A classical theorem of Steinitz [12, p. 125] states that the characteristic of an algebraically closed field, together with its absolute degree of transcendency, uniquely determine the field (up to isomorphism). It is easily seen that the word real-closed cannot be substituted for the words algebraically closed in this theorem. It is therefore natural to inquire what invariants other than the absolute transcendence degree are needed in order to characterize a real-closed field.' For non-denumerable fields, the question is equivalently stated as follows: what invariants in addition to the cardinal number are needed in order to characterize a real-closed field? Now, it is well-known that any two isomorphic realclosed fields are similarly ordered (i.e., as ordered sets). Here we establish the converse implication2 for a particular class of non-denumerable,3 non-archimedean, real-closed fields. Section 2 of our paper is devoted to the proof of this theorem (Theorem 2.1). The class of ordered fields to which our isomorphism theorem applies is quite restricted. (In fact, in order that it not be vacuous, we must assume either the continuum hypothesis, or some one of its generalizations to higher cardinals.4) Nevertheless, we are able to find an application to a class of fields that is not insignificant, namely, those that appear as residue class fields of maximal ideals in rings of continuous functions (on completely regular topological spaces). This discussion is the content of Section 3, and leads to the theorem that all nonarchimedean residue class fields (the so-called hyper-real fields) of power R, are isomorphic (Theorem 3.5). As a rather interesting corollary to this theorem, we find (using the continuum hypothesis) that all the non-real residue class fields of maximal ideals of a countable complete direct sum of real fields are isomorphic (Corollary 3.9). Section 4 continues the discussion of non-archimedean residue class fields. The development here leads to the construction of various such fields that arise from the same ring, but have different cardinal numbers (Theorems 4.4 ff. and 4.8 ff.). (A fortiori, not all such fields that arise from the same ring are isomorphic.) This section is almost entirely set-theoretic in character, and some of the results obtained here have some set-theoretic interest in themselves (Lemmas 4.1 and 4.7). (No use is made of the continuum hypothesis in this section.) Finally, in Section 5, we pose some unsolved problems.