  Using class field theory, we prove a restriction on the intersection of the
maximal abelian extensions associated with different number fields. This
restriction is then used to improve a result of Rosen and Silverman about the
linear independence of Heegner points. In addition, it yields effective
restrictions for the special points lying on an algebraic subvariety in a
product of modular curves. The latter application is related to the
Andr\'e-Oort conjecture.
