$3$-Principalization over $S_3$-fields

TL;DR

This paper investigates the capitulation of 3-ideal classes in certain pure metacyclic fields with Galois group S_3, analyzing their unramified extensions and the structure of their maximal unramified pro-3 extension.

Contribution

It determines the capitulation of 3-ideal classes in unramified cyclic cubic extensions for fields with specific 3-class group structure and explores implications for their maximal unramified pro-3 extensions.

Findings

Capitulation of 3-ideal classes is explicitly characterized.

Structure of the maximal unramified pro-3 extension is analyzed.

Results depend on the 3-class group type (9,3).

Abstract

Let $p\equiv 1\,(\mathrm{mod}\,9)$ be a prime number and $\zeta_3$ be a primitive cube root of unity. Then $\mathrm{k}=\mathbb{Q}(\sqrt[3]{p},\zeta_3)$ is a pure metacyclic field with group $\mathrm{Gal}(\mathrm{k}/\mathbb{Q})\simeq S_3$. In the case that $\mathrm{k}$ possesses a $3$-class group $C_{\mathrm{k},3}$ of type $(9,3)$, the capitulation of $3$-ideal classes of $\mathrm{k}$ in its unramified cyclic cubic extensions is determined, and conclusions concerning the maximal unramified pro-$3$-extension $\mathrm{k}_3^{(\infty)}$ of $\mathrm{k}$ are drawn.