  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 classify
CM-fields $k$ which are decomposed completely at $p$ such that
$X(\tilde{k})=0$.
