I will discuss intersections of class fields associated with distinct base fields and establish a lemma restricting their Galois groups. The proof of this lemma uses only standard class field theory. It should be also said that some special cases (related to ring class fields) are already in the literature (Cohn, Rosen-Silverman) and have suggested this general result. Then, I discuss applications of this new lemma to improve a result of Rosen and Silverman on the linear independence of Heegner points associated with distinct CM-fields. One can also use the lemma to prove some weak André-Oort type results on subvarieties in products of modular curves.