A note on extensions of ${\mathbb{Q}}^{tr}$

Abstract

In this note we investigate the behaviour of the absolute logarithmic Weil-height h on extensions of the field ${\mathbb{Q}}^{tr}$ of totally real numbers. It is known that there is a gap between zero and the next smallest value of h on ${\mathbb{Q}}^{tr}$, whereas in ${\mathbb{Q}}^{tr}(i)$ there are elements of arbitrarily small positive height. We prove that all elements of small height in any finite extension of ${\mathbb{Q}}^{tr}$ already lie in ${\mathbb{Q}}^{tr}(i)$. This leads to a positive answer to a question of Amoroso, David and Zannier, if there exists a pseudo algebraically closed field with the mentioned height gap.

References in corpus (2)

• Small Height and Infinite Non-Abelian Extensions
• On certain infinite extensions of the rationals with Northcott property