On fields with Property (B)

Amoroso, Francesco; David, Sinnou; Zannier, Umberto

doi:10.1090/S0002-9939-2014-11925-3

On fields with Property (B)
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by Francesco Amoroso, Sinnou David and Umberto Zannier

Proc. Amer. Math. Soc. 142 (2014), 1893-1910

DOI: https://doi.org/10.1090/S0002-9939-2014-11925-3

Published electronically: March 3, 2014

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Abstract:

Let $K$ be a number field and let $L/K$ be an infinite Galois extension with Galois group $G$. Let us assume that $G/Z(G)$ has finite exponent. We show that $L$ has the Property (B) of Bombieri and Zannier: the absolute and logarithmic Weil height on $L^*$ is bounded from below outside the set of roots of unity by an absolute constant. We also discuss some features of Property (B): stability by algebraic extensions and relations with field arithmetic. As a side result, we prove that the Galois group over $\mathbb {Q}$ of the compositum of all totally real fields is torsion free.

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