A note on small generators of number fields, II

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 $\textrm{Tor}\bigl(K^{\times}\bigr) = \{\pm 1\}$.
Comment: The statement and proof of Theorem 1.3 are corrected. We are grateful to Professor Hendrik Lenstra for pointing out an error in the previous draft