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In this paper we characterize weak multiplication modules.

Soient $d$ est un entier sans facteurs carrés, $\mathbf{K}=\mathbf{Q}(\sqrt{d},i)$, $i=\sqrt{-1}$, ${\mathbf{K}}_2^{\left(1\right)}$ le $2$-corps de classes de Hilbert de $\mathbf{K}$, ${\mathbf{K}}_2^{\left(2\right)}$ le $2$-corps de classes de Hilbert de ${\mathbf{K}}_2^{\left(1\right)}$ et $G=\mathrm{Gal}({\mathbf{K}}_2^{\left(2\right)}/\mathbf{K})$ le groupe de Galois de ${\mathbf{K}}_2^{\left(2\right)}/\mathbf{K}$. Notre but est de montrer qu’il existe une forme de $d$ tel que le $2$-groupe $G$ est non métacyclique et de donner une condition nécessaire et suffisante pour que le groupe $G$ soit métacyclique dans le cas où $d=2p$ avec $p$ un nombre premier tel que $p\equiv 1(mod4)$.

Soient $\mathbf{K}=\mathbf{Q}\left(\sqrt{-pq(2+\sqrt{2})}\right)$ où $p$ et $q$ deux nombres premiers différents tels que $p\equiv q\equiv \pm 5mod8$, ${\mathbf{K}}_2^{\left(1\right)}$ le $2$-corps de classes de Hilbert de $\mathbf{K}$, ${\mathbf{K}}_2^{\left(2\right)}$ le $2$-corps de classes de Hilbert de ${\mathbf{K}}_2^{\left(1\right)}$ et $G$ le groupe de Galois de ${\mathbf{K}}_2^{\left(2\right)}/K$. D’après [], la $2$-partie $C_{2,\mathbf{K}}$ du groupe de classes de $\mathbf{K}$ est de type $(2,2)$, par suite ${\mathbf{K}}_2^{\left(1\right)}$ contient trois extensions ${\mathbf{F}}_i/\mathbf{K}$ ; $i=1,2,3$. Dans ce papier, on s’interesse au problème de capitulation des $2$-classes d’idéaux de $\mathbf{K}$ dans ${\mathbf{F}}_i$ $(i=1,2,3)$ et à déterminer la structure de $G$.

It is well known by results of Golod and Shafarevich that the Hilbert $2$-class field tower of any real quadratic number field, in which the discriminant is not a sum of two squares and divisible by eight primes, is infinite. The aim of this article is to extend this result to any real abelian $2$-extension over the field of rational numbers. So using genus theory, units of biquadratic number fields and norm residue symbol, we prove that for every real abelian $2$-extension over $\mathbb{Q}$ in which eight primes...

Let $K$ be a biquadratic field, $K_2^{\left(1\right)}$ be the Hilbert $2$-class field of $K$ and $K_2^{\left(2\right)}$ be the Hilbert $2$-class field of $K_2^{\left(1\right)}$. Our goal is to prove that there exists a biquadratic field $K$ such that $Gal(K_2^{\left(1\right)}/K)\simeq \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ and the group $Gal(K_2^{\left(2\right)}/K)$ is semi-dihedral. Résumé. Soient $K$ un corps biquadratique, $K_2^{\left(1\right)}$ le $2$-corps de classes de Hilbert de $K$ et $K_2^{\left(2\right)}$ le $2$-corps de classes de Hilbert de $K_2^{\left(1\right)}$. Notre but est de prouver qu’il existe des corps biquadratiques réels $K$ tels que le groupe $Gal(K_2^{\left(1\right)}/K)$ est de type $(2,2)$ et le groupe $Gal(K_2^{\left(2\right)}/K)$ est semi-diédral.

Let $K=k\left(\sqrt{-p\epsilon \sqrt{l}}\right)$ with $k=\mathbb{Q}\left(\sqrt{l}\right)$ where $l$ is a prime number such that $l=2$ or $l\equiv 5mod8$, $\epsilon$ the fundamental unit of $k$, $p$ a prime number such that $p\equiv 1mod4$ and ${\left(\frac{p}{l}\right)}_4=-1$, $K_2^{\left(1\right)}$ the Hilbert $2$-class field of $K$, $K_2^{\left(2\right)}$ the Hilbert $2$-class field of $K_2^{\left(1\right)}$ and $G=Gal(K_2^{\left(2\right)}/K)$ the Galois group of $K_2^{\left(2\right)}/K$. According to E. Brown and C. J. Parry [7] and [8], $C_{2,K}$, the Sylow $2$-subgroup of the ideal class group of $K$, is isomorphic to $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$, consequently $K_2^{\left(1\right)}/K$ contains three extensions $F_i/K$ $(i=1,2,3)$ and the tower of the Hilbert $2$-class field of $K$ terminates at either $K_2^{\left(1\right)}$ or $K_2^{\left(2\right)}$. In this work, we are...

Let $K$ be an imaginary cyclic quartic number field whose 2-class group is of type $(2,2,2)$, i.e., isomorphic to $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$. The aim of this paper is to determine the structure of the Iwasawa module of the genus field $K^{(*)}$ of $K$.

Let G be some metabelian 2-group satisfying the condition G/G’ ≃ ℤ/2ℤ × ℤ/2ℤ × ℤ/2ℤ. In this paper, we construct all the subgroups of G of index 2 or 4, we give the abelianization types of these subgroups and we compute the kernel of the transfer map. Then we apply these results to study the capitulation problem for the 2-ideal classes of some fields k satisfying the condition $Gal(k{₂}^{\left(2\right)}/k)\simeq G$, where $k{₂}^{\left(2\right)}$ is the second Hilbert 2-class field of k.

Soient $p_1\equiv p_2\equiv 1(mod8)$ des nombres premiers tels que, $\left(\frac{p_1}{p_2}\right)=-1$ et $\left(\frac{2}{a+b}\right)=-1$, où $p_1{p}_2=a^2+b^2$. Soient $i=\sqrt{-1}$, $d=p_1{p}_2$, $𝕜=\mathbb{Q}(\sqrt{d},i)$, ${𝕜}_2^{\left(1\right)}$ le 2-corps de classes de Hilbert de $𝕜$ et ${𝕜}^{(*)}=\mathbb{Q}(\sqrt{p_1},\sqrt{p_2},i)$ le corps de genres de $𝕜$. La 2-partie $C_{𝕜,2}$ du groupe de classes de $𝕜$ est de type $(2,2,2)$, par suite ${𝕜}_2^{\left(1\right)}$ contient sept extensions quadratiques non ramifiées ${𝕂}_j/𝕜$ et sept extensions biquadratiques non ramifiées ${𝕃}_j/𝕜$. Dans ce papier on s’intéresse à déterminer ces quatorze extensions, le groupe $C_{𝕜,2}$ et à étudier la capitulation des 2-classes d’idéaux de $𝕜$ dans ces extensions.

We study the capitulation of $2$-ideal classes of an infinite family of imaginary bicyclic biquadratic number fields consisting of fields $𝕜=\mathbb{Q}(\sqrt{2pq},\mathrm{i})$, where $\mathrm{i}=\sqrt{-1}$ and $p\equiv -q\equiv 1(mod4)$ are different primes. For each of the three quadratic extensions $𝕂/𝕜$ inside the absolute genus field ${𝕜}^{(*)}$ of $𝕜$, we determine a fundamental system of units and then compute the capitulation kernel of $𝕂/𝕜$. The generators of the groups ${\mathrm{Am}}_s(𝕜/F)$ and $\mathrm{Am}(𝕜/F)$ are also determined from which we deduce that ${𝕜}^{(*)}$ is smaller than the relative genus field ${(𝕜/\mathbb{Q}\left(\mathrm{i}\right))}^{*}$. Then we prove that each...

Soient $k$ le corps quadratique réel $\mathbf{Q}\left(\sqrt{d}\right)$ (respectivement le corps biquadratique $\mathbf{Q}(\sqrt{d},\sqrt{-3})$), $d$ un entier positif sans facteur carré, $K$ une extension cubique cyclique non ramifiée de $k$, diédrale sur $\mathbf{Q}$ totalement réelle, (respectivement diédrale sur $\mathbf{Q}\left(\sqrt{-3}\right)$.) On constate qu’on a deux structures possibles pour le groupe des unités $U_K$ de $K$, notées $alpha$ et $delta$.

Let $d$ be a square free integer and $L_d:=\mathbb{Q}({\zeta }_8,\sqrt{d})$. In the present work we determine all the fields $L_d$ such that the $2$-class group, ${\mathrm{Cl}}_2\left(L_d\right)$, of $L_d$ is of type $(2,4)$ or $(2,2,2)$.

Let $d$ be an odd square-free integer, $m\ge 3$ any integer and $L_{m,d}:=\mathbb{Q}({\zeta }_{2^m},\sqrt{d})$. In this paper, we shall determine all the fields $L_{m,d}$ having an odd class number. Furthermore, using the cyclotomic ${\mathbb{Z}}_2$-extensions of some number fields, we compute the rank of the $2$-class group of $L_{m,d}$ whenever the prime divisors of $d$ are congruent to $3$ or $5(mod8)$.

Let $K=\mathbb{Q}\left(\theta \right)$ be a pure cubic field, with ${\theta }^3=D$, where $D$ is a cube-free integer. We will determine the reduced ideals of the order $𝒪=\mathbb{Z}\left[\theta \right]$ of $K$ which coincides with the maximal order of $K$ in the case where $D$ is square-free and $\neg \equiv \pm 1(mod9)$.

Let $𝕜=\mathbb{Q}\left(\sqrt{2},\sqrt{d}\right)$ be an imaginary bicyclic biquadratic number field, where $d$ is an odd negative square-free integer and ${𝕜}_2^{\left(2\right)}$ its second Hilbert $2$-class field. Denote by $G=\mathrm{Gal}({𝕜}_2^{\left(2\right)}/𝕜)$ the Galois group of ${𝕜}_2^{\left(2\right)}/𝕜$. The purpose of this note is to investigate the Hilbert $2$-class field tower of $𝕜$ and then deduce the structure of $G$.

Let $k$ be a number field with a 2-class group isomorphic to the Klein four-group. The aim of this paper is to give a characterization of capitulation types using group properties. Furthermore, as applications, we determine the structure of the second 2-class groups of some special Dirichlet fields $𝕜=\mathbb{Q}(\sqrt{d},\sqrt{-1})$, which leads to a correction of some parts in the main results of A. Azizi and A. Zekhini (2020).