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

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 ambiguous classes under Galois action.

Contribution

It provides a detailed analysis of capitulation phenomena in pure metacyclic fields of degree 20 with class group type (5,5), extending understanding of ideal class behavior in these fields.

Findings

Capitulation of 5-ideal classes is characterized for fields with class group (5,5).

All classes are ambiguous under the Galois group action in the studied cases.

Results contribute to the understanding of class group behavior in degree 20 metacyclic fields.

Abstract

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.