  For squarefree $d>1$, let $M$ denote the ring class field for the order
$Z[\sqrt{-3d}]$ in $F=Q(\sqrt{-3d})$. Hasse proved that $3$ divides the class
number of $F$ if and only if there exists a cubic extension $E$ of $Q$ such
that $E$ and $F$ have the same discriminant. Define the real cube roots
$v=(a+b\sqrt{d})^{1/3}$ and $v'=(a-b\sqrt{d})^{1/3}$, where $a+b\sqrt{d}$ is
the fundamental unit in $Q(\sqrt{d})$. We prove that $E$ can be taken as
$Q(v+v')$ if and only if $v \in M$. As byproducts of the proof, we give
explicit congruences for $a$ and $b$ which hold if and only if $v \in M$, and
we also show that the norm of the relative discriminant of $F(v)/F$ lies in
$\{1, 3^6\}$ or $\{3^8, 3^{18}\}$ according as $v \in M$ or $v \notin M$. We
then prove that $v$ is always in the ring class field for the order
$Z[\sqrt{-27d}]$ in $F$. Some of the results above are extended for subsets of
$Q(\sqrt{d})$ properly containing the fundamental units $a+b\sqrt{d}$.
