Geodesic
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, Tome 64 (2014) no. 1, pp. 11-29
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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.
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.
DOI : 10.1007/s10587-014-0078-9
Classification : 11R27, 11R29, 11R37
Mots-clés : unit group; class group; Hilbert class field; genus field; capitulation of ideal 
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     title = {Sur un probl\`eme de capitulation du corps $\mathbb {Q}(\sqrt {p_1p_2},\rm i)$ dont le $2$-groupe de classes est \'el\'ementaire},
     journal = {Czechoslovak Mathematical Journal},
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Azizi, Abdelmalek; Zekhnini, Abdelkader; 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, Tome 64 (2014) no. 1, pp. 11-29. doi: 10.1007/s10587-014-0078-9

[1] Azizi, A.: Units of certain imaginary abelian number fields over $\Bbb Q$. French Ann. Sci. Math. Qué. 23 15-21 (1999). | MR | Zbl

[2] Azizi, A.: Capitulation of the $2$-ideal classes of $\mathbb{Q}(\sqrt{p_1p_2}, \rm i )$ where $p_1$ and $p_2$ are primes such that $p_1\equiv 1 \pmod 8$, $p_2\equiv 5 \pmod 8$ and $(\frac{p_1}{p_2})=-1$. Algebra and Number Theory Boulagouaz, M'hammed et al. Proceedings of a conference, Fez, Morocco. Lect. Notes Pure Appl. Math. 208 Marcel Dekker, New York 13-19 (2000). | MR

[3] Azizi, A.: Construction of the 2-Hilbert class field tower of some biquadratic fields. French Pac. J. Math. 208 (2003), 1-10. | DOI | MR | Zbl

[4] Azizi, A.: On the units of certain number fields of degree 8 over $\Bbb Q$. Ann. Sci. Math. Qué. 29 (2005), 111-129. | MR | Zbl

[5] Azizi, A., Taous, M.: Determination of the fields $K=\Bbb Q(\sqrt d,\sqrt{-1})$, given the 2-class groups are of type $(2,4)$ or $(2,2,2)$. French. English summary Rend. Ist. Mat. Univ. Trieste 40 (2008), 93-116. | MR | Zbl

[6] Barruccand, P., Cohn, H.: Note on primes of type $x^2+32y^2$, class number, and residuacity. J. Reine Angew. Math. 238 (1969), 67-70. | MR

[7] Batut, C., Belabas, K., Bernadi, D., Cohen, H., Olivier, M.: GP/PARI calculator Version 2.2.6. (2003).

[8] Heider, F. P., Schmithals, B.: Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen. German J. Reine Angew. Math. 336 (1982), 1-25. | MR | Zbl

[9] Hilbert, D.: On the theory of the relative quadratic number field. Math. Ann. 51 (1899), 1-127.

[10] Kaplan, P.: Sur le $2$-groupe de classes d'idéaux des corps quadratiques. French J. Reine Angew. Math. 283/284 (1976), 313-363. | MR

[11] Lemmermeyer, F.: Reciprocity Laws. From Euler to Eisenstein. Springer Monographs in Mathematics, Springer, Berlin (2000). | MR | Zbl

[12] Parry, T. M. McCall. C. J., Ranalli, R. R.: Imaginary bicyclic biquadratic fields with cyclic $2$-class group. J. Number Theory 53 (1995), 88-99. | DOI | MR | Zbl

[13] Scholz, A.: Über die Lösbarkeit der Gleichung $t^2-Du^2=-4$. Math. Z. German 39 (1934), 95-111. | DOI

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