  Let $k \,=\, \mathbb{Q}(\sqrt[5]{n},\zeta_5)$, where $n$ is a positive
integer, $5^{th}$ power-free, whose $5-$class group is isomorphic to
$\mathbb{Z}/5\mathbb{Z}\times\mathbb{Z}/5\mathbb{Z}$. Let
$k_0\,=\,\mathbb{Q}(\zeta_5)$ be the cyclotomic field containing a primitive
$5^{th}$ root of unity $\zeta_5$. Let $C_{k,5}^{(\sigma)}$ the group of the
ambiguous classes under the action of $Gal(k/k_0)$ = $<\sigma>$. The aim of
this paper is to determine all integers $n$ such that the group of ambiguous
classes $C_{k,5}^{(\sigma)}$ has rank $1$ or $2$.
