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

TL;DR

This paper investigates the capitulation of 5-ideal classes in certain pure metacyclic fields of degree 20, focusing on fields with specific class group structures and their intermediate extensions.

Contribution

It provides new insights into the capitulation problem for pure metacyclic fields of degree 20 with particular class group configurations.

Findings

Capitulation behavior characterized for fields with class group (5,5).

Identified conditions under which ideal classes capitulate in intermediate extensions.

Enhanced understanding of class field theory in degree 20 metacyclic fields.

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