Capitulation des $2$-classes d’idéaux de $\mathbf{Q}\left(\sqrt{-pq\left(2+\sqrt{2}\right)}\right)$$p\equiv q\equiv ±5\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}8$
Annales Mathématiques Blaise Pascal, Tome 16 (2009) no. 1, pp. 57-69.

Soient $\mathbf{K}=\mathbf{Q}\left(\sqrt{-pq\left(2+\sqrt{2}\right)}\right)$$p$ et $q$ deux nombres premiers différents tels que $p\equiv q\equiv ±5\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}8$, ${\mathbf{K}}_{2}^{\left(1\right)}$ 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 [4], la $2$-partie ${C}_{2,\mathbf{K}}$ du groupe de classes de $\mathbf{K}$ est de type $\left(2,2\right)$, 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 ${\mathbf{F}}_{i}$ $\left(i=1,2,3\right)$ et à déterminer la structure de $G$.

Let $\mathbf{K}=\mathbf{Q}\left(\sqrt{-pq\left(2+\sqrt{2}\right)}\right)$ where $p$ and $q$ are two different prime numbers such that $p\equiv q\equiv ±5\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}8$, ${\mathbf{K}}_{2}^{\left(1\right)}$ the Hilbert $2$-class field of $\mathbf{K}$, ${\mathbf{K}}_{2}^{\left(2\right)}$ the Hilbert $2$-class field of ${\mathbf{K}}_{2}^{\left(1\right)}$ and $G$ the Galois group of ${\mathbf{K}}_{2}^{\left(2\right)}/\mathbf{K}$. According to [4], ${C}_{2,\mathbf{K}}$, the Sylow $2$-subgroup of the ideal class group of $\mathbf{K}$ is isomorphic to $\mathbf{Z}/2\mathbf{Z}×\mathbf{Z}/2\mathbf{Z}$, consequently ${\mathbf{K}}_{2}^{\left(1\right)}/\mathbf{K}$ contains three extensions ${\mathbf{F}}_{i}/\mathbf{K}$ $\left(i=1,2,3\right)$. In this paper, we are interested in the problem of capitulation of the classes of ${C}_{2,\mathbf{K}}$ in ${\mathbf{F}}_{i}$ $\left(i=1,2,3\right)$ and to determine the structure of $G$.

DOI : https://doi.org/10.5802/ambp.253
Classification : 11R27,  11R29,  11R37
Mots clés : Corps Quartiques, Groupes d’Unités, Corps de Classes de Hilbert, Capitulation
@article{AMBP_2009__16_1_57_0,
author = {Azizi, Abdelmalek and Talbi, Mohammed},
title = {Capitulation des $2$-classes d'id\'eaux de $\mathbf{Q}(\sqrt{-pq(2+\sqrt{2})})$ o\u $p\equiv q\equiv \pm 5\;\@mod \;8$},
journal = {Annales Math\'ematiques Blaise Pascal},
pages = {57--69},
publisher = {Annales math\'ematiques Blaise Pascal},
volume = {16},
number = {1},
year = {2009},
doi = {10.5802/ambp.253},
mrnumber = {2514527},
zbl = {1169.11046},
language = {fr},
url = {www.numdam.org/item/AMBP_2009__16_1_57_0/}
}
Azizi, Abdelmalek; Talbi, Mohammed. Capitulation des $2$-classes d’idéaux de $\mathbf{Q}(\sqrt{-pq(2+\sqrt{2})})$ où $p\equiv q\equiv \pm 5\;\@mod \;8$. Annales Mathématiques Blaise Pascal, Tome 16 (2009) no. 1, pp. 57-69. doi : 10.5802/ambp.253. http://www.numdam.org/item/AMBP_2009__16_1_57_0/`

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