Condition nécessaire et suffisante pour que certain groupe de Galois soit métacyclique

Azizi, Abdelmalek; Taous, Mohammed

Annales mathématiques Blaise Pascal, Tome 16 (2009), p. 83-92 / Harvested from Numdam

Access to full text
Full (PDF)
Full (DJVU)

Résumé
Abstract

Soient $d$ est un entier sans facteurs carrés, $K = Q (\sqrt{d}, i)$ , $i = \sqrt{- 1}$ , $K_{2}^{(1)}$ le $2$ -corps de classes de Hilbert de $K$ , $K_{2}^{(2)}$ le $2$ -corps de classes de Hilbert de $K_{2}^{(1)}$ et $G = Gal (K_{2}^{(2)} / K)$ le groupe de Galois de $K_{2}^{(2)} / K$ . Notre but est de montrer qu’il existe une forme de $d$ tel que le $2$ -groupe $G$ est non métacyclique et de donner une condition nécessaire et suffisante pour que le groupe $G$ soit métacyclique dans le cas où $d = 2 p$ avec $p$ un nombre premier tel que $p \equiv 1 (mod 4)$ .

Let $d$ be positive square-free integers, $K = Q (\sqrt{d}, i)$ and $i = \sqrt{- 1}$ . Let $K_{1}^{(2)}$ be the Hilbert $2$ -class field of $K$ , $K_{2}^{(2)}$ be the Hilbert $2$ -class field of $K_{1}^{(2)}$ and $G = Gal (K_{2}^{(2)} / K)$ be the Galois group of $K_{2}^{(2)} / K$ . Our goal is to show that there is some form of $d$ such $G$ is a nonmetacyclic $2$ -group and give the necessary condition and sufficient for the group $G$ to be metacyclic in case $d = 2 p$ with $p$ a prime number such that $p \equiv 1 (mod 4)$ .

Publié le : 2009-01-01

MR 2514529 | Zbl 1168.11046

DOI : https://doi.org/10.5802/ambp.255
Classification: 11R27, 11R29, 11R37
Mots clés: groupe des unités, système fondamentale d’unités, capitulation, corps de classes de Hilbert,

2

-groupe métacyclique

@article{AMBP_2009__16_1_83_0,
     author = {Azizi, Abdelmalek and Taous, Mohammed},
     title = {Condition n\'ecessaire et suffisante pour que certain groupe de Galois soit m\'etacyclique},
     journal = {Annales math\'ematiques Blaise Pascal},
     volume = {16},
     year = {2009},
     pages = {83-92},
     doi = {10.5802/ambp.255},
     zbl = {1168.11046},
     mrnumber = {2514529},
     language = {fr},
     url = {http://dml.mathdoc.fr/item/AMBP_2009__16_1_83_0}
}

Azizi, Abdelmalek; Taous, Mohammed. Condition nécessaire et suffisante pour que certain groupe de Galois soit métacyclique. Annales mathématiques Blaise Pascal, Tome 16 (2009) pp. 83-92. doi : 10.5802/ambp.255. http://gdmltest.u-ga.fr/item/AMBP_2009__16_1_83_0/

Bibliographie

[1] Azizi, A. Capitulation of the $2$ -ideal Classes of $ℚ (\sqrt{p_{1} p_{2}}, i)$ Where $p_{1}$ and $p_{2}$ are primes such that $p_{1} \equiv 1 (mod 8)$ , $p_{2} \equiv 5 (mod 8)$ and $(\frac{p_{1}}{p_{2}}) = - 1$ , Lecture notes in pure and applied mathematics, Tome 208 (1999), pp. 13-19 | MR 1724671 | Zbl 1003.11050

[2] Azizi, A. Sur une question de Capitulation, Proc. Amer. Math. Soc, Tome 130 (2002), pp. 2197-2202 | Article | MR 1897477 | Zbl 1010.11061

[3] Azizi, A.; Taous, M. Capitulation of $2$ -ideal classes of $k = ℚ (\sqrt{2 p}, i)$ in the genus field of $k$ where $p$ is prime such that $p \equiv 1 (mod 8)$ , IJPAM, Tome 35 (2007) no. 2, pp. 481-487 | MR 2311554 | Zbl pre05238028

[4] Baginski, C.; Konovalov, A. On $2$ -groups of almost maximal class, Publ. Math, Tome 65 (2004) no. 1-2, pp. 97-131 | MR 2075257 | Zbl 1070.20021

[5] Benjamin, E.; Lemmermeyer, F.; Snyder, C. Imaginary Quadratic Fields $k$ with Cyclic $C l_{2} (k^{1})$ , J. Number Theory, Tome 67 (1997), pp. 229-245 | Article | MR 1486501 | Zbl 0919.11074

[6] Benjamin, E.; Lemmermeyer, F.; Snyder, C. Real quadratic fields with abelian $2$ -class field tower, J. Number Theory, Tome 73 (1998), pp. 182-194 | Article | MR 1658015 | Zbl 0919.11073

[7] Benjamin, E.; Lemmermeyer, F.; Snyder, C. Imaginary quadratic fields with $C l_{2} (k) = (2, 2^{m})$ and rank $C l_{2} (k^{1}) = 2$ , Pac. J. Math, Tome 198 (2001), pp. 15-31 | Article | MR 1831970 | Zbl 1063.11038

[8] Benjamin, E.; Snyder, C. Number Fields with $2$ -class Number Isomorphic to $(2, 2^{m})$ (1994) (preprint)

[9] Blackburn, N. On Prime Power Groups in which the Derived Group has Two Generators, Proc. Cambridge Phil. Soc, Tome 53 (1957), pp. 19-27 | Article | MR 81904 | Zbl 0077.03202

[10] Blackburn, N. On a special class of $p$ -groups, Acta Math, Tome 100 (1958), pp. 45-92 | Article | MR 102558 | Zbl 0083.24802

[11] James, R. $2$ -Groups of Almost Maximal Class, J. Austral. Math. Soc. Ser. A, Tome 19 (1975), pp. 343-357 | Article | MR 382435 | Zbl 0309.20006

[12] Kisilevsky, H. Number fields with class number congruent to $4$ $\mod$ $8$ and Hilbert’s theorem $94$ , J. Number Theory, Tome 8 (1976) no. 3, pp. 271-279 | Article | MR 417128 | Zbl 0334.12019

[13] Kubota, T. Über die Beziehung der Klassenzahlen der Unterkörper des bizyklischen Zahlkörpers, Nagoya Math. J, Tome 6 (1953), pp. 119-127 | MR 59960 | Zbl 0053.21902

[14] Mccall, T.M.; Parry, C.J.; Ranalli, R.R. On imaginary bicyclic biquadratic fields with cyclic $2$ -class group, J. Number Theory, Tome 53 (1995), pp. 88-99 | Article | MR 1344833 | Zbl 0831.11059

[15] Taussky, O. A Remark on the Class Field Tower, J. Number Theory, Tome 12 (1937), pp. 82-85