Condition nécessaire et suffisante pour que certain groupe de Galois soit métacyclique

Azizi, Abdelmalek; Taous, Mohammed

Annales mathématiques Blaise Pascal, Tome 16 (2009) no. 1, pp. 83-92.

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 $p\equiv 1(mod4)$.

Let $d$ be positive square-free integers, $\mathbf{K}=\mathbf{Q}(\sqrt{d},i)$ and $i=\sqrt{-1}$. Let ${\mathbf{K}}_1^{\left(2\right)}$ be the Hilbert $2$-class field of $\mathbf{K}$, ${\mathbf{K}}_2^{\left(2\right)}$ be the Hilbert $2$-class field of ${\mathbf{K}}_1^{\left(2\right)}$ and $G=\mathrm{Gal}({\mathbf{K}}_2^{\left(2\right)}/\mathbf{K})$ be the Galois group of ${\mathbf{K}}_2^{\left(2\right)}/\mathbf{K}$. Our goal is to show that there is some form of $d$ such $G$ is a nonmetacyclic $2$-group and give the necessary condition and sufficient for the group $G$ to be metacyclic in case $d=2p$ with $p$ a prime number such that $p\equiv 1(mod4)$.

Classification : 11R27, 11R29, 11R37
Mots clés : groupe des unités, système fondamentale d’unités, capitulation, corps de classes de Hilbert, $2$-groupe métacyclique

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