  Let $k_\infty$ be the cyclotomic $\mathbb{Z}_p$-extension of an algebraic
number field $k$. We denote by $S$ a finite set of prime numbers which does not
contain $p$, and $S(k_\infty)$ the set of primes of $k_\infty$ lying above $S$.
In the present paper, we will study the structure of the Galois group
$\mathcal{X}_S (k_\infty)$ of the maximal pro-$p$ extension unramified outside
$S (k_\infty)$ over $k_\infty$. We mainly consider the question whether
$\mathcal{X}_S (k_\infty)$ is a non-abelian free pro-$p$ group or not. In the
former part, we treat the case when $k$ is an imaginary quadratic field and $S
= \emptyset$ (here $p$ is an odd prime number which does not split in $k$). In
the latter part, we treat the case when $k$ is a totally real field and $S \neq
\emptyset$.
