arXiv subjects

Fouad Elmouhib

Publications and source records attributed to Fouad Elmouhib.

6 recordsLinked to original sources

arXiv2022

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

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 $1$.

math.NT↗

arXiv2022

On the exact divisibility by $5$ of the class number of some pure metacyclic fields

Let $Γ\,=\, \mathbb{Q}(\sqrt[5]{n})$ be a pure quintic field, where $n$ is a natural number $5^{th}$ power-free. Let $k = \mathbb{Q}(\sqrt[5]{n}, ζ_5)$, with $ζ_5$ is a primitive $5^{th}$ root of unit, be the normal closure of $Γ$, and a pure metacyclic field of degree $20$ over $\mathbb{Q}$. When $n$ takes some particular forms, we show that $Γ$ admits a trivial $5$-class group and $5$ divides exactly the class number of $k$.

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arXiv2022

On some realizable metabelian $5$-groups

Let $G$ be a $5$-group of maximal class and $γ_2(G) = [G, G]$ its derived group. Assume that the abelianization $G/γ_2(G)$ is of type $(5, 5)$ and the transfers $V_{H_1\to γ_2(G)}$ and $V_{H_2\to γ_2(G)}$ are trivial, where $H_1$ and $H_2$ are two maximal normal subgroups of $G$. Then $G$ is completely determined with the isomorphism class groups of maximal class. Moreover the group $G$ is realizable with some fields $k$, which is the normal closure of a pure quintic field.

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arXiv2021

On the capitulation problem of some pure metacyclic fields of degree 20 II

Let $n$ be a $5^{th}$ power-free naturel number and $k_0\,=\,\mathbb{Q}(ζ_5)$ be the cyclotomic field generated by a primitive $5^{th}$ root of unity $ζ_5$. Then $k\,=\,\mathbb{Q}(\sqrt[5]{n},ζ_5)$ is a pure metacyclic field of absolute degree $20$. In the case that $k$ possesses a $5$-class group $C_{k,5}$ of type $(5,5)$ and all the classes are ambiguous under the action of $Gal(k/k_0)$, the capitulation of $5$-ideal classes of $k$ in its unramified cyclic quintic extensions is determined.

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arXiv2021

On the capitulation problem of some pure metacyclic fields of degree 20

Let $Γ\,=\, \mathbb{Q}(\sqrt[5]{n})$ be a pure quintic field, where $n$ is a positive integer $5^{th}$ power-free, $k_0\,=\,\mathbb{Q}(ζ_5)$ be the cyclotomic field containing a primitive $5^{th}$ root of unity $ζ_5$, and $k\,=\,\mathbb{Q}(\sqrt[5]{n},ζ_5)$ the normal closure of $Γ$. Let $k_5^{(1)}$ be the Hilbert $5$-class field of $k$, $C_{k,5}$ the $5$-ideal classes group of $k$, and $C_{k,5}^{(σ)}$ the group of ambiguous classes under the action of $Gal(k/k_0)$ = $\langleσ\rangle$. When $C_{k,5}$ is of type $(5,5)$ and rank $C_{k,5}^{(σ)}\,=\,1$, we study the capitulation problem of the $5$-ideal classes of $C_{k,5}$ in the six intermediate extensions of $k_5^{(1)}/k$.

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arXiv2020

$5$-rank of ambiguous class groups of quintic Kummer extensions

Let $k \,=\, \mathbb{Q}(\sqrt[5]{n},ζ_5)$, where $n$ is a positive integer, $5^{th}$ power-free, whose $5-$class group is isomorphic to $\mathbb{Z}/5\mathbb{Z}\times\mathbb{Z}/5\mathbb{Z}$. Let $k_0\,=\,\mathbb{Q}(ζ_5)$ be the cyclotomic field containing a primitive $5^{th}$ root of unity $ζ_5$. Let $C_{k,5}^{(σ)}$ the group of the ambiguous classes under the action of $Gal(k/k_0)$ = $<σ>$. The aim of this paper is to determine all integers $n$ such that the group of ambiguous classes $C_{k,5}^{(σ)}$ has rank $1$ or $2$.

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