Roots of unity and torsion points of abelian varieties

We answer a question raised by Hindry and Ratazzi concerning the intersection between cyclotomic extensions of a number field $K$ and extensions of $K$ generated by torsion points of an abelian variety over $K$. We prove that a weak version of the property $(\mu )$ they introduced holds for any abelian variety, while the same is not true for a slightly stronger version of the property they also considered.