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Let $\Gamma \,=\, \mathbb{Q}(\sqrt[5]{n})$ be a pure quintic field, where $n$is a positive integer, $5^{th}$ power-free. Let $k_0\,=\,\mathbb{Q}(\zeta_5)$be the cyclotomic field containing a primitive $5^{th}$ root of unity$\zeta_5$, and $k\,=\,\Gamma(\zeta_5)$ be the normal closure of $\Gamma$. Let$C_{k,5}$ be the $5$-component of the class group of k. The purpose of thispaper is to determine generators of $C_{k,5}$, whenever it is of type $(5,5)$and the rank of the group of ambiguous classes under the action of$Gal(k/k_0)\, =\,\langle \sigma\rangle$ is $1$.

The generators of $5$-class group of some fields of degree 20 over $\mathbb{Q}$