Small integral generators of totally complex number fields

TL;DR

This paper investigates the existence of small-height algebraic integers generating totally complex number fields, establishing conditions under which such generators exist or the field's structure constrains this possibility.

Contribution

The authors extend previous results by proving the existence of small-height generators in totally complex fields with specific Galois properties and characterize fields where such generators do not exist.

Findings

Positive answer for fields with nontrivial torsion in units.

Characterization of totally complex fields with trivial torsion as Galois over their real subfields.

Existence of small-height generators related to non-totally real algebraic integers.

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 $H(\alpha) \le H(\mu)\thinspace c_K$.