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The main purpose of this paper is to prove that the positive real numbers canbe decomposed into finitely many disjoint pieces which are also closed underaddition and multiplication.  As a byproduct of the argument we determine all the possible decompositionsof the transcendental extension of the rational field of rank one into twopieces. Further, we prove that the positive elements of a real algebraicextensions of the rational numbers are indecomposable into two pieces.

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