Adjunction of subfield closures to ordered division rings

Abstract

1. Preface. Let Z be an ordered division ring, P its prime field (i.e. the field of rationals), P* the closure of P in its order topology; P* is then orderisomorphic to the field of real numbers in their natural order. It has been shown by B. H. Neumann [1](1) that 2 can be extended to an ordered division ring Y4(P*) continuing the order of Z and containing P* in its centre. With a few changes in Neumann's proof the following generaliration will be proved: Let F be an arbitrary subfield of the centre of 1. Then z can be extended to an ordered diavision ring 2(F*) continuing the order of Z and containing the closure F* of F with respect to its order-topology in its centre. The new tool which is used throughout the paper is a mapping of ' into a system consisting of the symbols -X and +?X and an ordered residue class division ring D obtained by a peculiar type of fundamental sequence modulo the corresponding null-sequences. 2. The mapping u*5. Let 2 be an ordered di'vision ring with centre Z, F a fixed subfield of Z, and t+, F+ the multiplicative groups of the positive elements of 1, F respectively. We can assume the order of the additive semigroup of F to be nonarchimedean(2). Then the archimedean classes of the additive semigroups of I+' F+ form multiplicative groups a(z), a(F) when the product is defined by the product of representatives of the corresponding classes(3). The archimedean classes of the additive semigroups of 1+, F+ will be called additive archimedean classes of t+, F+ respectively. The additive archimedean class of an element ocG+ in a(T) will be denoted bv [1]. Thus [ar] consists of those rZ it can be constructed by means of fundamental sequences and null-sequences in the usual way. Thus F* is an ordered field containing (an ordered subfield isomorphic to) F such that its additive group is the order-topological completion of the additive group of F. \With this t-completion in the sense of CohenGoffman(4) a unique ordinal k* = t(F) is associated, and for the construction