Fields \(\mathbb{Q}(\sqrt[3]{d},\zeta_3)\) whose \(3\)-class group is of type \((9,3)\)

• Published in: arXiv.org
• Main Authors: Aouissi, Siham, Talbi, Mohamed, Ismaili, Moulay Chrif, Azizi, Abdelmalek
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 11.04.2019
• Subjects: Integers
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1805.04963

Let \(\mathrm{k}=\mathbb{Q}(\sqrt[3]{d},\zeta_3)\), with \(d\) a cube-free positive integer. Let \(C_{\mathrm{k},3}\) be the \(3\)-component of the class group of \(\mathrm{k}\). By the aid of genus theory, arithmetic proprieties of the pure cubic field \(\mathbb{Q}(\sqrt[3]{d})\) and some results on the \(3\)-class group \(C_{\mathrm{k},3}\), we are moving towards the determination of all integers \(d\) such that \(C_{\mathrm{k},3} \simeq \mathbb{Z}/9\mathbb{Z}\times \mathbb{Z}/3\mathbb{Z}\).