Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-04-30T23:49:00.435Z Has data issue: false hasContentIssue false
  • English
  • Français

Class-number problems for cubic number fields

Published online by Cambridge University Press:  22 January 2016

Stéphane Louboutin
Stéphane Louboutin*
Affiliation:
Université de Caen, U. F. R. SciencesDépartement de Mathématiques Esplanade de la Paix 14032 Caen Cedex, FRANCE email: loubouti@math.unicaen.fr
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let M be any number field. We let DM, dM, hu, , AM and RegM be the discriminant, the absolute value of the discriminant, the class-number, the Dedekind zeta-function, the ring of algebraic integers and the regulator of M, respectively.

we set If q is any odd prime we let (⋅/q) denote the Legendre’s symbol.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 138 , June 1995 , pp. 199 - 208
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1995

References

[Alb] Albert, A. A., A determination of the integers of all cubic fields, Ann. of Math., 31 (1930), 550–566.CrossRefGoogle Scholar
[Bar-Lou] Barrucand, B. and Louboutin, S., Majoration et minoration du nombre de classes d’idéaux des corps réels purs de degré premier, Bull. London Math. Soc, 25 (1993), 533–540.Google Scholar
[Bar-Loc-Will] Barrucand, P., Loxton, J. and Williams, H. C., Some explicit upper bounds on the class number and regulator of a cubic field with negative discriminant, Pacific J. of Math., 128 (1987), 209–222.Google Scholar
[Bar-Wil-Ban] Barrucand, P., Williams, H. C. and Baniuk, L., A computational technique for determining the class number of a pure cubic field, Math. Comp., 30 (1976), 312–323.CrossRefGoogle Scholar
[Cas-Fro] Cassels, J. W. S. and Fröhlich, A., Algebraic Number Theory, Chapter VIII Zeta-functions and L-functions, Academic Press, London, 1967.Google Scholar
[Fro-Tay] Fröhlich, A. and Taylor, M. J., Algebraic number theory, Cambridge Studies in Advanced Mathematics 27, Cambridge University Press 1991.Google Scholar
[Hos-Wad] Hosaya, H. and Wada, H., Tables of ideal class groups of purely cubic fields, Proc. Japan Acad., 68 (1992), 111–114.Google Scholar
[Hof] Hoffstein, J., On the siegel-Tatuzawa theorem, Acta Arith., 38 (1980), 167–174.CrossRefGoogle Scholar
[Laf] Lazarus, A. J., On the class number and unit index of simplest quartic fields, Nagoya Math. J., 121 (1991), 1–13.Google Scholar
[Let] Lettl, G., A lower bound for the class number of certain cubic number fields, Math. Comp., 46 (1986), 659–666.Google Scholar
[Lou 1] Louboutin, S., On the class number one problem for non-normal quartic CM-fields, Tohôku Math. J., 46 (1994), 1–12.Google Scholar
[Lou 2] Louboutin, S., Zéros réels des fonctions zêta et minorations de nombres de classes. Application à la majoration des discriminants de certains types de corps de nombres, Séminaire de Théorie des Nombres, Paris, 1991–92, Progress in Math. Vol. 116, 135–152, 1992 Birkhäuser.Google Scholar
[Mol-Wil] Mollin, R. A. and Williams, H. C., A conjecture of S. Chowla via the generalized Riemann hypothesis, Proc. Amer. Math. Soc, 102 (1988), 794–796.Google Scholar
[Nar] Narkiewicz, W., Elementary and Analytic Theory of Algebraic Numbers, Second Edition, Springer-Verlag, Berlin 1990.Google Scholar
[Sta] Stark, H. M., Some effective cases of the Brauer-Siegel theorem, Invent. Math., 23 (1974), 135–152.Google Scholar