On the triviality of the unramified Iwasawa modules of the maximal multiple \(\mathbb{Z}_p\)-extensions

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