Invariant Sets in Real Number Fields

Abstract

A set of complex numbers is called invariant if it is closed under addition and multiplication. We prove that each invariant set S containing a negative element, which is not in ${{\mathbb {Z}}}$, must be everywhere dense in ${{\mathbb {R}}}$. As in a recent work of Kiss, Somlai and Terpai, we are interested whether a given set can be decomposed into a disjoint union of several invariant sets. In this direction, we show that the intersection of a real number field K and a real interval I cannot be decomposed into a union of $m \ge 2$ disjoint invariant sets except for the trivial case $m=2$, $I=[0,+\infty )$, when $K \cap [0,+\infty )$ is the union of two disjoint invariant sets $\{0\}$ and $K^+$. In passing, we prove that, for each $a \in {{\mathbb {N}}}$ and each real number field K of degree $d \ge 2$, there is an algebraic integer $\beta \in K \cap (a,a+1)$ of degree d such that the coefficients of the minimal polynomial of $\beta$ in ${{\mathbb {Z}}}[x]$ are all positive except for its constant coefficient which is negative.