On the 2-class field tower of \mathbb Q (\sqrt{2p_1p_2},i) and the Galois group of its second Hilbert 2-class field

• Autores: Abdelmalek Azizi, Abdelkader Zekhnini, Mohammed Taous
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 65, Fasc. 1, 2014, págs. 131-141
• Idioma: inglés
• DOI: 10.1007/s13348-013-0085-4
• Texto completo no disponible (Saber más ...)
• Resumen

  • Let p_1 and p_2 be primes such that p_1\equiv p_2\equiv 5 \pmod 8, i=\sqrt{-1}, d=2p_1p_2, \mathbb K =\mathbb Q (\sqrt{d},i), \mathbb K _2^{(1)} be the Hilbert 2-class field of \mathbb K, \mathbb K _2^{(2)} be the Hilbert 2-class field of \mathbb K _2^{(1)}, G be the Galois group of \mathbb K _2^{(2)}/\mathbb K and \mathbb K ^{(*)}=\mathbb Q (\sqrt{p_1},\sqrt{p_2},\sqrt{2}, i) be the genus field of \mathbb K. The 2-part \mathbf C _{\mathbb{K },2} of the class group of \mathbb K is of type (2, 2, 2). Our goal is to study the 2-class field tower of \mathbb K and to calculate the order of G.

• Referencias bibliográficas
  • Azizi, A.: Unités de certains corps de nombres imaginaires et abéliens sur \mathbb{Q}. Ann. Sci. Math. Québec 23(1), 15–21 (1999)
  • Azizi, A.: Sur la capitulation des 2-classes d’idéaux de k =\mathbb{Q}(\sqrt{2pq}, i), où p\equiv -q\equiv 1 (\text{ mod }\; 4). Acta...
  • Azizi, A.: Sur les unités de certains corps de nombres de degré 8 sur \mathbb{Q}. Ann. Sci. Math. Québec 29(2), 111–129 (2005)
  • Azizi, A., Taous, M.: Détermination des corps {k} =\mathbb{Q}(\sqrt{d}, i) dont les 2-groupes de classes sont de type (2, 4) ou (2, 2,...
  • Azizi, A., Zekhnini, A., Taous, M.: On the unramified quadratic and biquadratic extensions of the field \mathbb{Q}(\sqrt{d}, i). IJA 6(24),...
  • Batut, C., Belabas, K., Bernadi, D., Cohen, H., Olivier, M.: GP/PARI calculator Version 2.2.6 (2003)
  • Blackburn, N.: On prime-power groups in which the derived group has two generators. Proc. Cambridge Phil. Soc. 53, 19–27 (1957)
  • Benjamin, E., Sanborn, F., Snyder, C.: Capitulation in unramified quadratic extensions of real quadratic number fields. Glasgow J. Math. 36,...
  • Benjamin, E., Lemmermeyer, F., Snyder, C.: Real quadratic fields with abelian 2-class field twoer. J. Number Theory 73(2), 182–194 (1998)
  • Gras, G.: Class field theory, from theory to practice, Springer, Berlin (2003)
  • Golod, E., Shafarevich, I.: On the class field tower. Izv. Akad. Nauk SSSR 28, 261–272 (1964) (in Russian); English translation: Am. Math....
  • Hirabayashi, M.: Unit indices of some imaginary composite quadratic fields II. Pacific J. Math. 173(1), 93–103 (1996)
  • Isaacs, I. M.: Character theory of finite groups. Academic Press, New York (1976)
  • Janko, Z.: A classification of finite 2-groups with exactly three involutions. J. Algebra 291, 505–533 (2005)
  • Kaplan, P.: Sur le 2-groupe de classes d’idéaux des corps quadratiques. J. Reine Angew. Math. 283(284), 313–363 (1976)
  • Koch, H.: Zum Satz von Golod-Schafarewitsch. Math. Nachr. 42, 321–333 (1968)
  • Kuroda, S.: Über den Dirichletschen Körper. J. Fac. Sci. Imp. Univ. Tokyo Sect. 14, 383–406 (1943)
  • Lemmermeyer, F.: Ideal class groups of cyclotomic number fields I. Acta Arithmetica LXXII.4 (1995)
  • Scholz, A.: Über die Lösbarkeit der Gleichung t^2-Du^2=-4. Math. Z. 39, 95–111 (1934)
  • Version 3, Copyright(C) 2007 free Software Foundation, Inc. (2007) http://fsf.org/
  • Wada, H.: On the class number and the unit group of certain algebraic number fields. J. Fac. Univ. Tokyo Sect. I 13, 201–209 (1966)