| Title:
|
Capitulation dans certaines extensions non ramifiées de corps quartiques cycliques (French) |
| Title:
|
Capitulation in certain nonramified extensions of cyclic quartic fields (English) |
| Author:
|
Azizi, Abdelmalek |
| Author:
|
Talbi, Mohammed |
| Language:
|
French |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
44 |
| Issue:
|
4 |
| Year:
|
2008 |
| Pages:
|
271-284 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $K=k\big (\sqrt{-p{\varepsilon }\sqrt{l}}\big )$ with $k={\mathbb{Q}}(\sqrt{l})$ where $l$ is a prime number such that $l=2$ or $l\equiv 5\;\@mod \;8$, $\varepsilon $ the fundamental unit of $k$, $p$ a prime number such that $p\equiv 1\;\@mod \;4$ and ${(\frac{p}{l})}_4=-1$, $K_2^{(1)}$ the Hilbert $2$-class field of $K$, $K_2^{(2)}$ the Hilbert $2$-class field of $K_2^{(1)}$ and $G=\operatorname{Gal\,}(K_2^{(2)}/K)$ the Galois group of $K_2^{(2)}/K$. 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 $K_2^{(1)}/K$ contains three extensions $F_i/K$ $(i=1,2,3)$ and the tower of the Hilbert $2$-class field of $K$ terminates at either $K_2^{(1)}$ or $K_2^{(2)}$. In this work, we are interested in the problem of capitulation of the classes of $C_{2,K}$ in $F_i$ $(i=1,2,3)$ and to determine the structure of $G$.
Résumé. Soient $K=k(\sqrt{-p{\varepsilon }\sqrt{l}})$ avec $k=\mathbb{Q}(\sqrt{l})$ où $l$ est un nombre premier tel que $l=2$ ou $l\equiv 5\;\@mod \;8$, $\varepsilon $ l’unité fondamentale de $k$, $p$ un nombre premier tels que $p\equiv 1\;\@mod \;4$ et ${(\frac{p}{l})}_4=-1$, $K_2^{(1)}$ le $2$-corps de classes de Hilbert de $K$, $K_2^{(2)}$ le $2$-corps de classes de Hilbert de $K_2^{(1)}$ et $G=\operatorname{Gal\,}(K_2^{(2)}/K)$ le groupe de Galois de $K_2^{(2)}/K$. D’après E. Brown et C. J. Parry [7] et [8], $C_{2,K}$, le $2$-groupe de classes de $K$, est isomorphe à $\mathbb{Z}/{2\mathbb{Z}}\times \mathbb{Z}/{2\mathbb{Z}}$, par conséquent $K_2^{(1)}/K$ contient trois extensions $F_i/K$ $(i=1,2,3)$ et la tour des $2$-corps de classes de Hilbert de $K$ s’arrête en $K_2^{(1)}$ ou en $K_2^{(2)}$. Dans ce travail, on s’intéresse au problème de capitulation des classes de $C_{2,K}$ dans $F_i$ $(i=1,2,3)$ et à déterminer la structure de $G$. (English) |
| Keyword:
|
corps biquadratiques cycliques |
| Keyword:
|
groupe de classes |
| Keyword:
|
capitulation |
| Keyword:
|
corps de classes de Hilbert |
| MSC:
|
11R27 |
| MSC:
|
11R37 |
| idZBL:
|
Zbl 1212.11091 |
| idMR:
|
MR2493424 |
| . |
| Date available:
|
2009-01-29T09:15:22Z |
| Last updated:
|
2013-09-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119767 |
| . |
| Reference:
|
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| Reference:
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