The generators of $3$-class group of some fields of degree $6$ over $\mathbb{Q}$

Abstract

Let $\mathrm{k}=\mathbb{Q}\left(\sqrt[3]{p},{\zeta }_3\right)$, where $p$ is a prime number such that $p\equiv 1(\mathrm{mod}9)$, and let $C_{\mathrm{k},3}$ be the $3$-component of the class group of $\mathrm{k}$. In \cite{GERTH3}, Frank Gerth III proves a conjecture made by Calegari and Emerton \cite{Cal-Emer} which gives necessary and sufficient conditions for $C_{\mathrm{k},3}$ to be of $\mathrm{rank}$ two. The purpose of the present work is to determine generators of $C_{\mathrm{k},3}$, whenever it is isomorphic to $\mathbb{Z}/9\mathbb{Z}\times \mathbb{Z}/3\mathbb{Z}$.

arXiv comment16 pages, 2 tables

Cited by in corpus (1)

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