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arXiv · 2005.04314

The generators of $5$-class group of some fields of degree 20 over $\mathbb{Q}$

Fouad Elmouhib·Mohamed Talbi·Abdelmalek Azizi
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Abstract

Let $Γ\,=\, \mathbb{Q}(\sqrt[5]{n})$ be a pure quintic field, where $n$ is a positive integer, $5^{th}$ power-free. Let $k_0\,=\,\mathbb{Q}(ζ_5)$ be the cyclotomic field containing a primitive $5^{th}$ root of unity $ζ_5$, and $k\,=\,Γ(ζ_5)$ be the normal closure of $Γ$. Let $C_{k,5}$ be the $5$-component of the class group of k. The purpose of this paper is to determine generators of $C_{k,5}$, whenever it is of type $(5,5)$ and the rank of the group of ambiguous classes under the action of $Gal(k/k_0)\, =\,\langle σ\rangle$ is $1$.

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Fouad Elmouhib, Mohamed Talbi, Abdelmalek Azizi. 2022-08-16. The generators of $5$-class group of some fields of degree 20 over $\mathbb{Q}$. https://arxiv.org/abs/2005.04314

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