Small generators of abelian number fields

We show that for each abelian number field $K$ of sufficiently large degree $d$ there exists an element $\alpha \in K$ with $K=\text{IQ}(\alpha )$ and absolute Weil height $H(\alpha ){\ll }_d\mathrm{\mid }{\mathrm{\Delta }}_K{\mathrm{\mid }}^{1\mathrm{/}2d}$ , where ${\mathrm{\Delta }}_K$ 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 $1\mathrm{/}2d$ is best-possible when $d$ is even.