arXiv subjects

Siham Aouissi

Publications and source records attributed to Siham Aouissi.

11 recordsLinked to original sources

arXiv2025

Coclass of the second 3-class group

By means of parametrized presentations of finite metabelian 3-groups, it is proved that the coclass cc(M) of the second 3-class group M=Gal(F_3^2(K)/K) of any algebraic number field K with elementary bicyclic 3-class group Cl_3(K)=(3,3) is determined unambiguously by the second largest order ord(Cl_3(E_2))=3^{cc(M)+1} among the four 3-class groups of the unramified cyclic cubic extensions E_i (i=1,..,4) of K. Minimal discriminants of quadratic and cubic fields K with assigned coclass cc(M) are computed from extensive databases of 3-class numbers ord(Cl_3(E_i)) as an application.

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arXiv2025

The Capitulation Problem in Certain Pure Cubic Fields

Let $Γ=\mathbb{Q}(\sqrt[3]{n})$ be a pure cubic field with normal closure $k=\mathbb{Q}(\sqrt[3]{n},ζ)$, where $n>1$ denotes a cube free integer, and $ζ$ is a primitive cube root of unity. Suppose $k$ possesses an elementary bicyclic $3$-class group $\mathrm{Cl}_3(k)$, and the conductor of $k/\mathbb{Q}(ζ)$ has the shape $f\in\lbrace pq_1q_2,3pq,9pq\rbrace$ where $p\equiv 1\,(\mathrm{mod}\,9)$ and $q,q_1,q_2\equiv 2,5\,(\mathrm{mod}\,9)$ are primes. It is disproved that there are only two possible capitulation types $\varkappa(k)$, either type $\mathrm{a}.1$, $(0000)$, or type $\mathrm{a}.2$, $(1000)$. Evidence is provided, theoretically and experimentally, of two further types, $\mathrm{b}.10$, $(0320)$, and $\mathrm{d}.23$, $(1320)$.

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arXiv2024

Group theory of cyclic cubic number fields

Astonishing new discoveries with quartets and octets of cyclic cubic fields sharing a common conductor are presented. Four kinds of graphs describing cubic residue conditions among the prime divisors of the conductor enforce elementary bi- or tricyclic 3-class groups and either a metabelian 3-class field tower group of coclass at least two or a closed Andozhskii-Tsvetkov group of order 6561. In the latter situation, abelian type invariants of first and second order are required for the identification.

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arXiv2024

Group theoretic approach to cyclic cubic fields

Let (k1,k2,k3,k4) be a quartet of cyclic cubic number fields sharing a common conductor c=pqr divisible by exactly three prime(power)s p,q,r. For those components k of the quartet whose 3-class group Cl(3,k) = Z/3Z x Z/3Z is elementary bicyclic, the automorphism group M = Gal(F(3,2,k)/k) of the maximal metabelian unramified 3-extension of k is determined by conditions for cubic residue symbols between p,q,r and for ambiguous principal ideals in subfields of the common absolute 3-genus field k* of k1,k2,k3,k4. With the aid of the relation rank d2(M), it is decided whether M coincides with the Galois group G = Gal(F(3,infinity,k)/k) of the maximal unramified pro-3-extension of k.

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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.

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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.

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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.

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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 $C_{\mathrm{k},3} \simeq \mathbb{Z}/9\mathbb{Z}\times \mathbb{Z}/3\mathbb{Z}$.

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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.

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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.

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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 $\mathbb{Z}/9\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$.

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