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arXiv · 1808.04678

Structure of relative genus fields of cubic Kummer extensions

Siham Aouissi·Abdelmalek Azizi·Moulay Chrif Ismaili·Daniel C. Mayer·Mohamed Talbi
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Abstract

Let $N=K(\sqrt[3]{D})$ be a cubic Kummer extension of the cyclotomic field $K=\mathbb{Q}(\zeta_3)$, containing a primitive cube root of unity $\zeta_3$, with cube free integer radicand $D>1$. Denote by $f$ the conductor of the abelian extension $N/K$, and by $N^{\ast}$ the relative genus field of $N/K$. The aim of the present work is to find out all positive integers $D$ and conductors $f$ such that the genus group $\operatorname{Gal}\left(N^{\ast}/N\right)\cong \mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}$ is elementary bicyclic.

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Siham Aouissi, Abdelmalek Azizi, Moulay Chrif Ismaili, Daniel C. Mayer, Mohamed Talbi. 2018-08-14. Structure of relative genus fields of cubic Kummer extensions. https://doi.org/10.1007/s40590-023-00562-8

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