  Let $K$ be an algebraic number field and $H$ the absolute Weil height. Write
$c_K$ for a certain positive constant that is an invariant of $K$. We consider
the question: does $K$ contain an algebraic integer $\alpha$ such that both $K
= \mathbb{Q}(\alpha)$ and $H(\alpha) \le c_K$? If $K$ has a real embedding then
a positive answer was established in previous work. Here we obtain a positive
answer if $\textrm{Tor}\bigl(K^{\times}\bigr) \not= \{\pm 1\}$, and so $K$ has
only complex embeddings. We also show that if the answer is negative, then $K$
is totally complex, $K$ has a subfield $F$ that is totally real, $K/F$ is a
Galois extension, and $\textrm{Tor}\bigl(K^{\times}\bigr) = \{\pm 1\}$.
