arXiv subjects

Mohamed Talbi

Publications and source records attributed to Mohamed Talbi.

14 recordsLinked to original sources

arXiv2023

Structure of relative genus fields of cubic Kummer extensions

Let $N=K(\sqrt[3]{D})$ be a cubic Kummer extension of the cyclotomic field $K=\mathbb{Q}(ζ_3)$, containing a primitive cube root of unity $ζ_3$, 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.

math.NT↗

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 $k$.

math.NT↗

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.

math.NT↗

arXiv2021

$3$-Principalization over $S_3$-fields

Let $p\equiv 1\,(\mathrm{mod}\,9)$ be a prime number and $ζ_3$ be a primitive cube root of unity. Then $\mathrm{k}=\mathbb{Q}(\sqrt[3]{p},ζ_3)$ is a pure metacyclic field with group $\mathrm{Gal}(\mathrm{k}/\mathbb{Q})\simeq S_3$. In the case that $\mathrm{k}$ possesses a $3$-class group $C_{\mathrm{k},3}$ of type $(9,3)$, the capitulation of $3$-ideal classes of $\mathrm{k}$ in its unramified cyclic cubic extensions is determined, and conclusions concerning the maximal unramified pro-$3$-extension $\mathrm{k}_3^{(\infty)}$ of $\mathrm{k}$ are drawn.

math.NT↗

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 $k_5^{(1)}/k$.

math.NT↗

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 $2$.

math.NT↗

arXiv2019

5-Class towers of cyclic quartic fields arising from quintic reflection

Let zeta5 be a primitive fifth root of unity and d<>1 be a quadratic fundamental discriminant not divisible by 5. For the 5-dual cyclic quartic field M=Q((zeta5-zeta5^-1)*d^1/2) of the quadratic fields k1=Q(d^1/2) and k2=Q((5*d)^1/2) in the sense of the quintic reflection theorem, the possibilities for the isomophism type of the Galois group G(5,2)M=Gal(M(5,2)/M) of the second Hilbert 5-class field M(5,2) of M are investigated, when the 5-class group Cl5(M) is elementary bicyclic of rank two. Usually, the maximal unramified pro-5-extension M(5,infinity) of M coincides with M(5,2) already. The precise length ell5(M) of the 5-class tower of M is determined, when G(5,2)M is of order less than or equal to 5^5. Theoretical results are underpinned by the actual computation of all 83, respectively 93, cases in the range 0<d<10^4, respectively -2*10^5<d<0.

math.NT↗

arXiv2019

Principal factors and lattice minima

Let $\mathit{k}=\mathbb{Q}(\sqrt[3]{d},ζ_3)$, where $d>1$ is a cube-free positive integer, $\mathit{k}_0=\mathbb{Q}(ζ_3)$ be the cyclotomic field containing a primitive cube root of unity $ζ_3$, and $G=\operatorname{Gal}(\mathit{k}/\mathit{k}_0)$. The possible prime factorizations of $d$ in our main result [2, Thm. 1.1] give rise to new phenomena concerning the chain $Θ=(θ_i)_{i\in\mathbb{Z}}$ of \textit{lattice minima} in the underlying pure cubic subfield $L=\mathbb{Q}(\sqrt[3]{d})$ of $\mathit{k}$. The aims of the present work are to give criteria for the occurrence of generators of primitive ambiguous principal ideals $(α)\in\mathcal{P}_{\mathit{k}}^G/\mathcal{P}_{\mathit{k}_0}$ among the lattice minima $Θ=(θ_i)_{i\in\mathbb{Z}}$ of the underlying pure cubic field $L=\mathbb{Q}(\sqrt[3]{d})$, and to explain exceptional behavior of the chain $Θ$ for certain radicands $d$ with impact on determining the principal factorization type of $L$ and $\mathit{k}$ by means of Voronoi's algorithm.

math.NT↗

arXiv2019

Fields $\mathbb{Q}(\sqrt[3]{d},ζ_3)$ whose $3$-class group is of type $(9,3)$

Let $\mathrm{k}=\mathbb{Q}(\sqrt[3]{d},ζ_3)$, 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 $C_{\mathrm{k},3} \simeq \mathbb{Z}/9\mathbb{Z}\times \mathbb{Z}/3\mathbb{Z}$.

math.NT↗

arXiv2019

$3$-rank of ambiguous class groups of cubic Kummer extensions

Let $k=k_0(\sqrt[3]{d})$ be a cubic Kummer extension of $k_0=\mathbb{Q}(ζ_3)$ with $d>1$ a cube-free integer and $ζ_3$ a primitive third root of unity. Denote by $C_{k,3}^{(σ)}$ the $3$-group of ambiguous classes of the extension $k/k_0$ with relative group $G=\operatorname{Gal}(k/k_0)=\langleσ\rangle$. The aims of this paper are to characterize all extensions $k/k_0$ with cyclic $3$-group of ambiguous classes $C_{k,3}^{(σ)}$ of order $3$, to investigate the multiplicity $m(f)$ of the conductors $f$ of these abelian extensions $k/k_0$, and to classify the fields $k$ according to the cohomology of their unit groups $E_{k}$ as Galois modules over $G$. The techniques employed for reaching these goals are relative $3$-genus fields, Hilbert norm residue symbols, quadratic $3$-ring class groups modulo $f$, the Herbrand quotient of $E_{k}$, and central orthogonal idempotents. All theoretical achievements are underpinned by extensive computational results.

math.NT↗

arXiv2019

On a Conjecture of Lemmermeyer

Let $p\equiv 1\,(\mathrm{mod}\,3)$ be a prime and denote by $ζ_3$ a primitive third root of unity. Recently, Lemmermeyer presented a conjecture about $3$-class groups of pure cubic fields $L=\mathbb{Q}(\sqrt[3]{p})$ and of their normal closures $\mathrm{k}=\mathbb{Q}(\sqrt[3]{p},ζ_3)$. The purpose of this paper is to prove Lemmermeyer's conjecture.

math.NT↗

arXiv2018

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

Let $\mathrm{k}=\mathbb{Q}\left(\sqrt[3]{p},ζ_3\right)$, where $p$ is a prime number such that $p \equiv 1 \pmod 9$, 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 $\operatorname{rank}\,$ two. The purpose of the present work is to determine generators of $C_{\mathrm{k},3}$, whenever it is isomorphic to $\mathbb{Z}/9\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$.

math.NT↗

arXiv2017

Self-generated off-axis holography in interferometric out-of-focus imaging of ice particles

Interferometric out-of-focus imaging of ice particles is realized. The overlapping defocused images of two nearby ice particles are analyzed. If one particle is much smaller than the other one, the pattern constitutes an off-axis hologram whose analysis gives the exact size and shape of the biggest particle. Using two angles of view, we obtain a 3D description of the biggest ice particle.

physics.optics↗