Real closures of fields at orderings of higher level

Abstract

Natural maps are defined here which allow many questions about £. Becker's of to be reduced to questions about the value groups of the places they induce. A simple construction is given of the set of orderings of level which induce a given place (this set is bijective with the set of subgroups of the value group of the place whose factor groups are cyclic of 2-power order). This construction leads to straightforward valuation-theoretic characterizations of real closed fields and of real closures of fields at orderings of The sets of isomorphism classes of real closures of a field which induce a given place, a given ordering of any level, or even a given family of orderings are each explicitly computed. 1. Introduction. Throughout this paper, will denote a field. An ordering of level (abbreviated: ordering) of is a subset of which is maximal with respect to exclusion of -1 and closure under addition and multiplication (i.e., a Harrison prime) which contains F2 for some n [B]. If is such an ordering, then T= T {0} is a subgroup of the multiplicative group F'= F {0} and F'/T is cyclic of order 2m for some integer m (m is called the exact level of T) [B]. Following Lam [L, §12], we will call the orderings of exact level one ordinary orderings; these are precisely the orderings appearing in the classical Artin-Schreier theory of formally real fields. Orderings of exact level greater than one will be called orderings of higher exact level. Orderings of level have received considerable attention since being introduced by Becker in 1978 [B, Bl, BHR, Cr, H, KR]. Our exposition, while inevitably overlapping earlier work, is largely independent of it and is at least to some extent distinguished from it by an increased emphasis on natural constructions and mappings. Our starting point is the fact, essentially due to Harrison and Warner [HW, Theorem 1.1] and independently rediscovered by Becker [B, p. 18], that each ordering is associated in a canonical way with a place into the field R of real numbers, i.e., with a real-valued place.