On fields with the Property (B)

International audienceLet 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