On fields with the Property (B)

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^* (outside the set of roots of unity) is bounded from below by an absolute constant. We discuss some feature of Property (B): stability by algebraic extensions, relations with field arithmetic. As a as a side result, we prove that the Galois group over Q of the compositum of all totally real fields is torsion free.
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Francesco Amoroso, Sinnou David, Umberto Zannier. On fields with the Property (B). Proceedings of the American Mathematical Society, American Mathematical Society, 2014, 142 (6), pp.1893-1910. ⟨hal-00649954⟩

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