Small integral generators of totally complex number fields

Abstract

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, $\textrm{Tor}\bigl(K^{\times}\bigr) = \{\pm 1\}$, and $K$ is a Galois extension of its maximal totally real subfield. Further, we show that if $\mu \in O_K$ is not totally real, then there exists $\alpha$ in $O_K$ with $K = \mathbb{Q}(\alpha)$ and $H(\alpha) \le H(\mu)\thinspace c_K$.