“…On the other hand, by arguments similar to the proof of Iwasawa's p-divisibility theorem ( [4], see also [13,Theorem 10.4]), Theorem 1.2 for k with cyclic A(k) is also obtained under more general ramification condition (see [14, Theorem 2.1 and Corollary 2.3], also [11,Proposition 1]). In this paper, combining the ideas of these results, we give a generalization of Theorem 1.2 in more general situation where the number fields are not necessarily finite extensions of Q.…”