  Let $K$ be a field finitely generated over the field of rational numbers,
$K(c)$ the extension of $K$ obtained by adjoining all roots of unity, $L$ an
infinite Galois extension of $K$, $X$ an abelian variety defined over $K$. We
prove that under certain conditions on $X$ and $K$ the existence of infinitely
many L-rational points of finite order on $X$ implies that the intersection of
$L$ and $K(c)$ has infinite degree over $K$.
