$3$-Principalization over $S_3$-fields

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.

23 pages, 7 Figures, 1 Table