A construction of an ordered division ring with a rank one valuation

Ka Hin Leung

Vol. 147 (1991), No. 1, 139–151

DOI: 10.2140/pjm.1991.147.139

Abstract

Let (k,P_k) be an ordered field and Γ be a dense additive subgroup of ℝ. In this paper, we shall construct a noncommutative ordered division ring (D,P) and a compatible valuation v on (D,P) such that (i) the value group of v is Γ and (ii) the residue division ring (D_v,P_v) is order isomorphic to (k,P_k). This problem is interesting because, in effect, we are constructing the “simplest” or in some sense the smallest noncommutative ordered division ring.

Mathematical Subject Classification 2000

Primary: 16W80

Secondary: 12E15, 12J15

Milestones

Received: 26 January 1989

Revised: 22 October 1989

Published: 1 January 1991