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Tables of octic fields with a quartic subfield


Authors: H. Cohen, F. Diaz y Diaz and M. Olivier
Journal: Math. Comp. 68 (1999), 1701-1716
MSC (1991): Primary 11R37, 11Y40
DOI: https://doi.org/10.1090/S0025-5718-99-01074-1
Published electronically: February 24, 1999
MathSciNet review: 1642813
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Abstract: We describe the computation of extended tables of degree 8 fields with a quartic subfield, using class field theory. In particular we find the minimum discriminants for all signatures and for all the possible Galois groups. We also discuss some phenomena and statistics discovered while making the tables, such as the occurrence of 11 non-isomorphic number fields having the same discriminant, or several pairs of non-isomorphic number fields having the same Dedekind zeta function.


References [Enhancements On Off] (What's this?)

  • [Be] K. Belabas, A fast algorithm to compute cubic fields, Math. Comp. 66 (1997), 1213-1237. MR 97m:11159
  • [Bu-Mc] G. Butler and J. McKay, The transitive groups of degree up to eleven, Comm. Algebra 11 (1983), 863-911. MR 84f:20005
  • [Co] H. Cohen, A course in computational algebraic number theory, GTM 138, Springer-Verlag, Berlin, Heidelberg, New York, 1993. MR 94i:11105
  • [Co-Di] H. Cohen and F. Diaz y Diaz, A polynomial reduction algorithm, Sém. Th. des Nombres Bordeaux (série 2) 3 (1991), 351-360. MR 93a:11107
  • [Co-Di-Ol] H. Cohen, F. Diaz y Diaz and M. Olivier, Computing ray class groups, conductors and discriminants, Math. Comp. 67 (1998), 773-795. MR 98g:11128
  • [Co-Hu-Mc] J. Conway, A. Hulpke and J. McKay, On transitive permutation groups, LMS J. Comput. Math., 1 (1998), 1-8.
  • [Di-Ol] F. Diaz y Diaz and M. Olivier, Imprimitive ninth-degree number fields with small discriminants, Math. Comp. 64 (1995), 305-321. MR 95e:11153
  • [Ei] Y. Eichenlaub, Problèmes effectifs de théorie de Galois en degrés $8$ à $11$, Thèse, Université Bordeaux I, 1996.
  • [Ei-Ol] Y. Eichenlaub and M. Olivier, Computation of Galois groups for polynomials with degree up to eleven (1996), Preprint.
  • [Ga] F. Gassmann, Bemerkungen zur Vorstehenden Arbeit von Hurwitz, Math. Z. 25 (1926), 665-675.
  • [Ha] H. Hasse, Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper, Physica-Verlag, Wien-Würtzburg, 1970. MR 42:1795
  • [Je] A. Jehanne, Sur les extensions de $\mathbb Q$ à groupe de Galois $S_4$ et $\widetilde S_4$, Acta Arith. 69 (1995), 259-276. MR 95m:11126
  • [Kw] S.-H. Kwon, Sur les discriminants minimaux des corps quaternioniens, Arch. Math. 67 (1996), 119-125. MR 97d:11165
  • [Pe] R. Perlis, On the equation $\zeta _K(s)=\zeta _{K'}(s)$, J. Number Th. 9 (1977), 342-360. MR 56:5503
  • [Po-Za] M. Pohst and H. Zassenhaus, Algorithmic algebraic number theory, Encyclopedia of Math. and its Applications, Cambridge University Press, Cambridge, 1989. MR 92b:11074
  • [Ro] X.-F. Roblot, Unités de Stark et corps de classes de Hilbert, C. R. Acad. Sci. Paris 323 (1996), 1165-1168. MR 97k:11162
  • [Sm-Pe] B. de Smit and R. Perlis, Zeta functions do not determine class numbers, Bull. Am. Math. Soc. 31 (1994), 213-215. MR 95a:11100
  • [Sm] G. W. Smith, Some polynomials over $\mathbb Q(t)$ and their Galois groups (1993), Preprint.

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Additional Information

H. Cohen
Affiliation: Laboratoire A2X, Université Bordeaux I, 351 cours de la Libération, 33405 Talence Cedex, France
Email: cohen@math.u-bordeaux.fr

F. Diaz y Diaz
Affiliation: Laboratoire A2X, Université Bordeaux I, 351 cours de la Libération, 33405 Talence Cedex, France
Email: diaz@math.u-bordeaux.fr

M. Olivier
Affiliation: Laboratoire A2X, Université Bordeaux I, 351 cours de la Libération, 33405 Talence Cedex, France
Email: olivier@math.u-bordeaux.fr

DOI: https://doi.org/10.1090/S0025-5718-99-01074-1
Keywords: Class field theory, discriminant, number field
Received by editor(s): November 20, 1997
Published electronically: February 24, 1999
Article copyright: © Copyright 1999 American Mathematical Society

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