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In this note we investigate the behaviour of the absolute logarithmic Weil-height h on extensions of the field ℚtr of totally real numbers. It is known that there is a gap between 0 and the next smallest value of h on ℚtr, whereas in ℚtr(i) there are elements of arbitrarily small positive height. We prove that all elements of small height in any finite extension of ℚtr already lie in ℚtr(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.
