On fields with Property (B)

Abstract

Let K K be a number field and let L / K L/K be an infinite Galois extension with Galois group G G . Let us assume that G / Z ( G ) G/Z(G) has finite exponent. We show that L L has the Property (B) of Bombieri and Zannier: the absolute and logarithmic Weil height on L ∗ 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 Q \mathbb {Q} of the compositum of all totally real fields is torsion free.