| Title:
|
Sur un problème de capitulation du corps $\mathbb {Q}(\sqrt {p_1p_2},\rm i)$ dont le $2$-groupe de classes est élémentaire (French) |
| Title:
|
On a capitulation problem over the field $\mathbb {Q}(\sqrt {p_1p_2},\rm i)$ with elementary $2$-class group (English) |
| Author:
|
Azizi, Abdelmalek |
| Author:
|
Zekhnini, Abdelkader |
| Author:
|
Taous, Mohammed |
| Language:
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French |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
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1572-9141 (online) |
| Volume:
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64 |
| Issue:
|
1 |
| Year:
|
2014 |
| Pages:
|
11-29 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $p_1\equiv p_2\equiv 1 \pmod 8$ be primes such that $(\frac {p_1}{p_2})=-1$ and $(\frac {2}{a+b})=-1$, where $p_1p_2=a^2+b^2$. Let ${\rm i}=\sqrt {-1}$, $d=p_1p_2$, $\Bbbk =\mathbb {Q}(\sqrt {d},{\rm i})$, $\Bbbk _2^{(1)}$ be the Hilbert 2-class field and $\Bbbk ^{(*)}=\mathbb{Q} (\sqrt {p_1},\sqrt {p_2},{\rm i})$ be the genus field of $\Bbbk $. The 2-part ${\bf C}_{{\Bbbk },2}$ of the class group of $\Bbbk $ is of type $(2,2,2)$, so $\Bbbk _2^{(1)}$ contains seven unramified quadratic extensions $\mathbb K_j/\Bbbk $ and seven unramified biquadratic extensions $\mathbb {L}_j/\Bbbk $. Our goal is to determine the fourteen extensions, the group ${\bf C}_{{\Bbbk },2}$ and to study the capitulation problem of the 2-classes of $\Bbbk $. \medskip {\it Résumé. Soient $p_1\equiv p_2\equiv 1\pmod 8$ des nombres premiers tels que, $(\frac {p_1}{p_2})=-1$ et $(\frac {2}{a+b})=-1$, où $p_1p_2=a^2+b^2$. Soient ${\rm i}=\sqrt {-1}$, $d=p_1p_2$, $\Bbbk =\mathbb {Q}(\sqrt {d},{\rm i})$, $\Bbbk _2^{(1)}$ le 2-corps de classes de Hilbert de $\Bbbk $ et $\Bbbk ^{(*)}=\mathbb{Q} (\sqrt {p_1},\sqrt {p_2},{\rm i})$ le corps de genres de $\Bbbk $. La 2-partie ${\bf C}_{{\Bbbk },2}$ du groupe de classes de $\Bbbk $ est de type $(2, 2, 2)$, par suite $\Bbbk _2^{(1)}$ contient sept extensions quadratiques non ramifiées $\mathbb K_j/\Bbbk $ et sept extensions biquadratiques non ramifiées $\mathbb {L}_j/\Bbbk $. Dans ce papier on s'intéresse à déterminer ces quatorze extensions, le groupe ${\bf C}_{{\Bbbk },2}$ et à étudier la capitulation des 2-classes d'idéaux de $\Bbbk $ dans ces extensions. (English) |
| Summary:
|
Soient $p_1\equiv p_2\equiv1\pmod8$ des nombres premiers tels que, $(\frac{p_1}{p_2})=-1$ et $(\frac2{a+b})=-1$, où $p_1p_2=a^2+b^2$. Soient $ i=\sqrt{-1}$, $d=p_1p_2$, $\Bbbk=\mathbb{Q}(\sqrt{d}, i)$, $\Bbbk_2^{(1)}$ le 2-corps de classes de Hilbert de $\Bbbk$ et $\Bbbk^{(*)}=\mathbb Q(\sqrt{p_1},\sqrt{p_2}, i)$ le corps de genres de $\Bbbk$. La 2-partie $ C_{{\Bbbk},2}$ du groupe de classes de $\Bbbk$ est de type $(2, 2, 2)$, par suite $\Bbbk_2^{(1)}$ contient sept extensions quadratiques non ramifiées $\mathbb K_j/\Bbbk$ et sept extensions biquadratiques non ramifiées $\mathbb{L}_j/\Bbbk$. Dans ce papier on s'intéresse à déterminer ces quatorze extensions, le groupe $ C_{{\Bbbk},2}$ et à étudier la capitulation des 2-classes d'idéaux de $\Bbbk$ dans ces extensions. (French) |
| Keyword:
|
unit group |
| Keyword:
|
class group |
| Keyword:
|
Hilbert class field |
| Keyword:
|
genus field |
| Keyword:
|
capitulation of ideal |
| MSC:
|
11R27 |
| MSC:
|
11R29 |
| MSC:
|
11R37 |
| idZBL:
|
Zbl 06391471 |
| idMR:
|
MR3247439 |
| DOI:
|
10.1007/s10587-014-0078-9 |
| . |
| Date available:
|
2014-09-29T09:28:10Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143944 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
|
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| Reference:
|
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| Reference:
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