Every Real Closed Field Has an Integer Part

Let us call an integer part of an ordered field any subring such that every element of the field lies at distance less than I from a unique element of the ring. We show that every real closed field has an integer part. ?1. Definitions, remarks. 1.1. Let Kdenote the real closure of the totally ordered field K. 1.2. We say that a subring Z of a ring A is an integer part of A if it is discrete and if for any x E A, there is z E Z such that z < x < z + 1. Then we call this unique element z the integer part of x and we write z = [x]. Shepherdson [0] showed that a discrete abelian ring is a model of Open Induction iff it is the integer part of its real closure. His construction, based on Puiseux series, is generalized here by using transfinite series (introduced by Mal'cev and B. von Neumann). Since the result reported here was obtained, integer parts of ordered fields have turned out to be a useful tool for the study of fields. 1.3. S. Boughattas showed in [1] that on the one hand, every totally ordered field has an ultrapower endowed with an integer part, and on the other hand, that there are ordered fields without an integer part. In fact, he has a p-real closed field with no integer part for every integer p. We show these examples to be optimal, in the sense that every real closed field has an integer part. 1.4. In fact we will show a stronger result: let A be a convex subring of K = K-; then any integer part Z of A can be extended to an integer part ZK of K. The proof uses the convex valuation on K defined by A. ?2. Convex valuation in a totally ordered field. The following results are classical results of valuation theory (cf., [2, 3]). Let K be a (totally) ordered field, w a convex valuation on K; we denote by k the residue field and F = w(K) the abelian totally ordered value group. When y E K and w(y) = 0, jwill be the residue image of y in k. 2.1. PROPOSITION. If K is real closed, then k is real closed and F divisible. Moreover k can be embedded in K and there exists a cross section, i.e., a family Received February 3, 1992; revised April 29, 1992. ? 1993, Association for Symbolic Logic 0022-4812/93/5802-001 i/$01 .70