标题： 正实数的分解为在加法和乘法下封闭的不相交集合 显示英文标题

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

摘要： 本文的主要目的是证明正实数可以分解为有限多个不相交的块，这些块在加法和乘法下也是封闭的。 作为论证的副产品，我们确定了有理数域的一阶超越扩张分解为两块的所有可能方式。 此外，我们证明了有理数的实代数扩张中的正元素无法分解为两块。 显示英文摘要

摘要： The main purpose of this paper is to prove that the positive real numbers can be decomposed into finitely many disjoint pieces which are also closed under addition and multiplication. As a byproduct of the argument we determine all the possible decompositions of the transcendental extension of the rational field of rank one into two pieces. Further, we prove that the positive elements of a real algebraic extensions of the rational numbers are indecomposable into two pieces.