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

TL;DR

This paper studies the behavior of the Weil-height on extensions of totally real numbers, showing that small-height elements in finite extensions are contained in a specific quadratic extension, answering a question about height gaps.

Contribution

It proves that all small-height elements in finite extensions of $ ext{Q}^{tr}$ are contained in $ ext{Q}^{tr}(i)$, confirming a conjecture related to height gaps and algebraic fields.

Findings

Small height elements in finite extensions of $ ext{Q}^{tr}$ lie in $ ext{Q}^{tr}(i)

Confirms the existence of a height gap in these fields

Answers a question about pseudo algebraically closed fields and height gaps

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.