An Isomorphism Theorem for Real-Closed Fields

A classical theorem of Steinitz [I& p. 1251 states that the characteristic of an algebraically closed field, together with it.s absolute degree of transcendency, uniquely det,ermine 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 charact’erize a real-closed field? Xow, it is well-known that any two isomorphic realclosed fields are similarly ordered (i.e., as ordered s&s). Here we establish the converse implication’ 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 funcbions (on completely regular topological spaces). This discussion is the content of Section 3, and leads t’o the theorem that all nonarchimedean residue class fields (the so-called kyper-reul fields) of power N1 are isomorphic (Theorem 3.5). As a rat,her interesting corollary to this theorem, we find (using the continuum hypot’hesis) that all the non-real residue class fields of maximal ideals of a countable complete direct sum of real fields are isomorphic (Corolla~ry 3.9). Section 4 continues the discussion of non-arahimedean 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.). (it 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 t,he results obtained here have some set-theoretic interest in themselves (Lemmas 4.1 and 4.7). (No use is made of the continuum hypot’hesis in this section.) Finally, in Section 5, we pose some unsolved problems.