Characterization of the numbers which satisfy the height reducing property

Abstract

Let $\alpha$ be a complex number. We show that there is a finite subset $F$ of the ring of the rational integers $\mathbb{Z}$, such that $F\left[\alpha \right]=\mathbb{Z}\left[\alpha \right]$, if and only if $\alpha$ is an algebraic number whose conjugates, over the field of the rationals, are all of modulus one, or all of modulus greater than one. This completes the answer to a question, on the numbers satisfying the height reducing property, posed in [3].

Indagationes Mathematicae (2014)