Let $S$ be a non-empty set of prime numbers; $1 \leq |S| \leq \infty$.
Let $\mathbf{Q}^S$ denote the abelian extension of
the rational field $\mathbf{Q}$ whose Galois group over $\mathbf{Q}$
is topologically isomorphic to the direct product of the additive
groups of $l$-adic integers for all $l \in S$. In this note,
we shall give simple examples of $S$ such that, for some $l \in S$,
the Hilbert $l$-class field over $\mathbf{Q}^S$ is
a nontrivial extension of $\mathbf{Q}^S$. Our results imply
that, if $S$ contains 2, 3, 31, and 73, then there exists an
unramified cyclic extension of degree $2263 = 31 \cdot 73$
over $\mathbf{Q}^S$.