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Let $k \,=\, \mathbb{Q}(\sqrt[5]{n},\zeta_5)$, where $n$ is a positiveinteger, $5^{th}$ 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 theambiguous classes under the action of $Gal(k/k_0)$ = $<\sigma>$. The aim ofthis paper is to determine all integers $n$ such that the group of ambiguousclasses $C_{k,5}^{(\sigma)}$ has rank $1$ or $2$.

$5$-rank of ambiguous class groups of quintic Kummer extensions