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 $k_5^{ (1)}/k$.
Categories: math.NT