On the capitulation problem of some pure metacyclic fields of degree 20

Abstract

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 $Gal(k/k_0)$ = $⟨\sigma ⟩$. 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 $k_5^{(1)}/k$.

10 pages, 1 figures