  Let $\Gamma \,=\, \mathbb{Q}(\sqrt[5]{n})$ be a pure quintic field, where $n$
is a positive integer $5^{th}$ power-free, $k_0\,=\,\mathbb{Q}(\zeta_5)$ be the
cyclotomic field containing a primitive $5^{th}$ root of unity $\zeta_5$, and
$k\,=\,\mathbb{Q}(\sqrt[5]{n},\zeta_5)$ the normal closure of $\Gamma$. Let
$k_5^{(1)}$ be the Hilbert $5$-class field of $k$, $C_{k,5}$ the $5$-ideal
classes group of $k$, and $C_{k,5}^{(\sigma)}$ the group of ambiguous classes
under the action of $Gal(k/k_0)$ = $\langle\sigma\rangle$. When $C_{k,5}$ is of
type $(5,5)$ and rank $C_{k,5}^{(\sigma)}\,=\,1$, we study the capitulation
problem of the $5$-ideal classes of $C_{k,5}$ in the six intermediate
extensions of $k_5^{(1)}/k$.
