On the triviality of the unramified Iwasawa modules of the maximal multiple Zₚ-extensions

For a number field k and an odd prime number p, we consider the maximal multiple Zₚ-extension k of k and the unramified Iwasawa module X (k), which is the Galois group of the maximal unramified abelian p-extension over k. In the present article, we classify CM-fields k which are decomposed completely at p such that X (k) =0.