Small generators of abelian number fields

Abstract

We show that for each abelian number field $K$ of sufficiently large degree $d$ there exists an element $\alpha \in K$ with $K=\IQ(α)$ and absolute Weil height $H(\alpha ){\ll }_d|{\Delta }_K{|}^{1/2d}$ , where ${\Delta }_K$ denotes the discriminant of $K$. This answers a question of Ruppert from 1998 in the case of abelian extensions of sufficiently large degree. We also show that the exponent $1/2d$ is best-possible when $d$ is even.