Sur le $2$-groupe de classes des corps multiquadratiques réels

Mouhib, Ali; Movahhedi, Abbas

Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 2, pp. 619-641.

Soient $p_1,p_2,...,p_n$ des nombres premiers distincts $\not\equiv -1\left(mod4\right)$, $d:=p_1{p}_2\cdots p_n$ et $k_n=\mathbf{Q}(\sqrt{p_1},\sqrt{p_2},...,\sqrt{p_n})$. On peut approcher le $2$-rang du groupe de classes des corps $k_n$ en étudiant celui du corps $k_m\left(\sqrt{d}\right)$ pour un entier $m<n$. Dans cet article, on traite le cas où $m=2$ ou $3$. Comme application, on déduit que le rang du $2$-groupe de classes de $k_4$ est au moins égal à deux (on savait déjà grâce à un résultat de Fröhlich que le groupe de classes de $k_4$ est toujours d’ordre pair). On en déduit également la liste de tous les corps multiquadratiques $k_n$ ayant un $2$-groupe de classes cyclique non trivial.

Let $p_1,p_2,...,p_n$ be distinct rational prime numbers $\not\equiv -1\left(mod4\right)$, $d:=p_1{p}_2\cdots p_n$ and $k_n=\mathbf{Q}(\sqrt{p_1},\sqrt{p_2},...,\sqrt{p_n})$. The $2$-rank of the class group of $k_n$ can be approached by studying that of the field $k_m\left(\sqrt{d}\right)$, for an integer $m<n$. In this article, we treat the case where $m=2$ or $3$. As an application, we deduce that the rank of the $2$-class group of $k_4$ is at least two (according to a result of Fröhlich, we already knew that the class group of $k_4$ is always of even order). We also draw the list of all multiquadratic fields $k_n$ whose $2$-class group is cyclic non-trivial.

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