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A note on heights in certain infinite extensions of \(\mathbb Q\). (English) Zbl 1072.11077
Let \(h(\alpha)\) denote the absolute logarithmic Weil height of an algebraic number \(\alpha\). A set \(A\) of algebraic numbers is said to have the Northcott property if, for every \(T > 0\), the set of all \(\alpha \in A\) with \(h(\alpha) < T\) is finite. D. G. Northcott [Proc. Camb. Philos. Soc. 45, 502–509 (1949; Zbl 0035.30701)] proved that the set of all algebraic numbers of degree at most \(d\) has this property.
In this article, the authors prove that the compositum of all quadratic extensions of \(\mathbb Q\) has the Northcott property. They also investigate, for certain infinite extensions of \(\mathbb Q\), the Bogomolov property, which is said to hold for a set \(A\) of algebraic numbers if there exists a \(T_0 > 0\) such that the set of all \(\alpha \in A\) with \(h(\alpha) < T_0\) is exactly the set of roots of unity in \(A\).

MSC:
11R04 Algebraic numbers; rings of algebraic integers
11G50 Heights
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