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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
##### Keywords:
algebraic number theory; heights; uniform distribution
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