On a capitulation problem over the field ℚ ( p 1 p 2 , 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

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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 $p_1\equiv p_2\equiv 1\hspace\{4.44443pt\}(\@mod \; 8)$ 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 $ i=\sqrt\{-1\}$, $d=p_1p_2$, $\mathbb \{k\}=\mathbb \{Q\}(\sqrt\{d\}, i)$, $\mathbb \{k\}_2^\{(1)\}$ le 2-corps de classes de Hilbert de $\mathbb \{k\}$ et $\mathbb \{k\}^\{(*)\}=\mathbb \{Q\}(\sqrt\{p_1\},\sqrt\{p_2\}, i)$ le corps de genres de $\mathbb \{k\}$. La 2-partie $ C_\{\{\mathbb \{k\}\},2\}$ du groupe de classes de $\mathbb \{k\}$ est de type $(2, 2, 2)$, par suite $\mathbb \{k\}_2^\{(1)\}$ contient sept extensions quadratiques non ramifiées $\mathbb \{K\}_j/\mathbb \{k\}$ et sept extensions biquadratiques non ramifiées $\mathbb \{L\}_j/\mathbb \{k\}$. Dans ce papier on s’intéresse à déterminer ces quatorze extensions, le groupe $ C_\{\{\mathbb \{k\}\},2\}$ et à étudier la capitulation des 2-classes d’idéaux de $\mathbb \{k\}$ dans ces extensions.},
author = {Azizi, Abdelmalek, Zekhnini, Abdelkader, Taous, Mohammed},
journal = {Czechoslovak Mathematical Journal},
keywords = {unit group; class group; Hilbert class field; genus field; capitulation of ideal; 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 $p_1\equiv p_2\equiv 1\hspace{4.44443pt}(\@mod \; 8)$ 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 $ i=\sqrt{-1}$, $d=p_1p_2$, $\mathbb {k}=\mathbb {Q}(\sqrt{d}, i)$, $\mathbb {k}_2^{(1)}$ le 2-corps de classes de Hilbert de $\mathbb {k}$ et $\mathbb {k}^{(*)}=\mathbb {Q}(\sqrt{p_1},\sqrt{p_2}, i)$ le corps de genres de $\mathbb {k}$. La 2-partie $ C_{{\mathbb {k}},2}$ du groupe de classes de $\mathbb {k}$ est de type $(2, 2, 2)$, par suite $\mathbb {k}_2^{(1)}$ contient sept extensions quadratiques non ramifiées $\mathbb {K}_j/\mathbb {k}$ et sept extensions biquadratiques non ramifiées $\mathbb {L}_j/\mathbb {k}$. Dans ce papier on s’intéresse à déterminer ces quatorze extensions, le groupe $ C_{{\mathbb {k}},2}$ et à étudier la capitulation des 2-classes d’idéaux de $\mathbb {k}$ dans ces extensions.
LA - fre
KW - unit group; class group; Hilbert class field; genus field; capitulation of ideal; capitulation; biquadratic field; unit group; class group; Hilbert class field; genus field
UR - http://eudml.org/doc/262006
ER -

References

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