Decompositions of the positive real numbers into disjoint sets closed under addition and multiplication

Abstract

The general problem studied in this paper is to decompose the positive elements R+of a totally ordered ring Rinto the disjoint union of sets that are closed under addition and multiplication. We mainly investigate the case when R is a subfield of the reals R. We prove that for any n ∈Nthe positive real numbers can be decomposed into n disjoint pieces which are also closed under addition and multiplication. We construct infinitely many different n-decompositions (n ∈N) for all fields containing at least two algebraically independent transcendental numbers. We characterise all possible decompositions into two pieces for fields of transcendence degree one over Q. Further, we prove that the positive elements of real algebraic extensions of the rational numbers are indecomposable into finitely many pieces.