  Let $p\equiv 1\,(\mathrm{mod}\,9)$ be a prime number and $\zeta_3$ be a
primitive cube root of unity. Then $\mathrm{k}=\mathbb{Q}(\sqrt[3]{p},\zeta_3)$
is a pure metacyclic field with group
$\mathrm{Gal}(\mathrm{k}/\mathbb{Q})\simeq S_3$. In the case that $\mathrm{k}$
possesses a $3$-class group $C_{\mathrm{k},3}$ of type $(9,3)$, the
capitulation of $3$-ideal classes of $\mathrm{k}$ in its unramified cyclic
cubic extensions is determined, and conclusions concerning the maximal
unramified pro-$3$-extension $\mathrm{k}_3^{(\infty)}$ of $\mathrm{k}$ are
drawn.

Number Theory