  We show that for each abelian number field $K$ of sufficiently large degree
$d$ there exists an element $\alpha\in K$ with $K=\IQ(\alpha)$ 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.
