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Let $\Gamma \,=\, \mathbb{Q}(\sqrt[5]{n})$ be a pure quintic field, where $n$ is a natural number $5^{th}$ power-free. Let $k = \mathbb{Q}(\sqrt[5]{n}, \zeta_5)$, with $\zeta_5$ is a primitive $5^{th}$ root of unit, be the normal closure of $\Gamma$, and a pure metacyclic field of degree $20$ over $\mathbb{Q}$. When $n$ takes some particular forms, we show that $\Gamma$ admits a trivial $5$-class group and $5$ divides exactly the class number of $k$., Comment: 10 pages, 3 tables

Let $n$ be a $5^{th}$ power-free naturel number and $k_0\,=\,\mathbb{Q}(\zeta_5)$ be the cyclotomic field generated by a primitive $5^{th}$ root of unity $\zeta_5$. Then $k\,=\,\mathbb{Q}(\sqrt[5]{n},\zeta_5)$ is a pure metacyclic field of absolute degree $20$. In the case that $k$ possesses a $5$-class group $C_{k,5}$ of type $(5,5)$ and all the classes are ambiguous under the action of $Gal(k/k_0)$, the capitulation of $5$-ideal classes of $k$ in its unramified cyclic quintic extensions is determined., Comment: 12 pages

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 $k_5^{(1)}/k$., Comment: 10 pages, 1 figures