  Let $D>1$ be an integer, and let $b=b(D)>1$ be its smallest divisor. We show
that there are infinitely many number fields of degree $D$ whose primitive
elements all have relatively large height in terms of $b$, $D$ and the
discriminant of the number field. This provides a negative answer to a
questions of W. Ruppert from 1998 in the case when $D$ is composite.
Conditional on a very weak form of a folk conjecture about the distribution of
number fields, we negatively answer Ruppert's question for all $D>3$.
