On a capitulation problem over the field $\mathbb {Q}(\sqrt {p_1p_2},\rm i)$ with elementary $2$-class group

Abdelmalek Azizi; Abdelkader Zekhnini; Mohammed Taous

Czechoslovak Mathematical Journal (2014)

  • Volume: 64, Issue: 1, page 11-29
  • ISSN: 0011-4642

Abstract

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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.

How to cite

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Azizi, Abdelmalek, Zekhnini, Abdelkader, and Taous, Mohammed. "Sur un problème de capitulation du corps $\mathbb {Q}(\sqrt {p_1p_2},\rm i)$ dont le $2$-groupe de classes est élémentaire." Czechoslovak Mathematical Journal 64.1 (2014): 11-29. <http://eudml.org/doc/262006>.

@article{Azizi2014,
abstract = {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.},
author = {Azizi, Abdelmalek, Zekhnini, Abdelkader, Taous, Mohammed},
journal = {Czechoslovak Mathematical Journal},
keywords = {capitulation; biquadratic field; unit group; class group; Hilbert class field; genus field},
language = {fre},
number = {1},
pages = {11-29},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Sur un problème de capitulation du corps $\mathbb \{Q\}(\sqrt \{p_1p_2\},\rm i)$ dont le $2$-groupe de classes est élémentaire},
url = {http://eudml.org/doc/262006},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Azizi, Abdelmalek
AU - Zekhnini, Abdelkader
AU - Taous, Mohammed
TI - Sur un problème de capitulation du corps $\mathbb {Q}(\sqrt {p_1p_2},\rm i)$ dont le $2$-groupe de classes est élémentaire
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 1
SP - 11
EP - 29
AB - 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.
LA - fre
KW - capitulation; biquadratic field; unit group; class group; Hilbert class field; genus field
UR - http://eudml.org/doc/262006
ER -

References

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