  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.
