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

Abstract

Let $n$ be a $5^{th}$ power-free natural 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.