  We establish upper bounds for the smallest height of a generator of a number
field $k$ over the rational field $\Q$. Our first bound applies to all number
fields $k$ having at least one real embedding. We also give a second
conditional result for all number fields $k$ such that the Dedekind
zeta-function associated to the Galois closure of $k/\Q$ satisfies GRH. This
provides a partial answer to a question of W. Ruppert.
