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The class number one problem for some
non-abelian normal CM-fields


Authors: Stéphane Louboutin, Ryotaro Okazaki and Michel Olivier
Journal: Trans. Amer. Math. Soc. 349 (1997), 3657-3678
MSC (1991): Primary 11R29; Secondary 11R21, 11R42, 11M20, 11Y40
DOI: https://doi.org/10.1090/S0002-9947-97-01768-6
MathSciNet review: 1390044
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Abstract | References | Similar Articles | Additional Information

Abstract: Let ${\bf N}$ be a non-abelian normal CM-field of degree $4p,$ $p$ any odd prime. Note that the Galois group of ${\bf N}$ is either the dicyclic group of order $4p,$ or the dihedral group of order $4p.$ We prove that the (relative) class number of a dicyclic CM-field of degree $4p$ is always greater then one. Then, we determine all the dihedral CM-fields of degree $12$ with class number one: there are exactly nine such CM-fields.


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

  • [BP] E. Brown and C.J. Parry. The imaginary bicyclic biquadratic fields with class number $1.$ J. Reine Angew. Math. 266 (1974), 118-120. MR 49:4974
  • [BWW] D.A. Buell, H.C. Williams and K.S. Williams. On the imaginary bicyclic biquadratic fields with class number $2.$ Math. Comp. 31 (1977), 1034-1042.MR 56:305
  • [Coh] H. Cohen. A course in computational algebraic number theory. Springer-Verlag, Grad. Texts Math. 138, 1993. MR 94i:11105
  • [Has] H. Hasse. Über die Klassenzahl abelscher Zahlkörper. Akademie-Verlag, Berlin 1952.MR 14:141a
  • [Hof] J. Hoffstein. Some analytic bounds for zeta functions and class numbers. Inventiones Math. 55 (1979), 37-47.MR 80k:12019
  • [Hor] K. Horie. On a ratio between relative class number. Math. Z. 211 (1992), 505-521.MR 94a:11171
  • [JY] C.U. Jensen and N. Yui. Polynomials with $D_p$ as Galois group. J. Number Th. 15 (1982), 347-375.MR 84g:12011
  • [Lou 1] S. Louboutin. Continued fractions and real quadratic fields. J. Number Theory 30 (1988), 167-176.MR 90a:11119
  • [Lou 2] S. Louboutin. Norme relative de l'unité fondamentale et $2$-rang du groupe des classes d'idéaux de certains corps biquadratiques. Acta Arith. 58 (1991), 273-288.MR 93a:11090
  • [Lou 3] S. Louboutin. Calcul des nombres de classes relatifs de certains corps de classes de Hilbert. C. R. Acad. Sci. Paris Sér. I Math. 319 I (1994), 321-325.MR 95g:11111
  • [Lou 4] S. Louboutin. Calcul du nombre de classes des corps de nombres. Pacific J. Math. 17 (1995), 455-467.MR 97a:11176
  • [Lou 5] S. Louboutin. Lower bounds for relative class numbers of CM-fields. Proc. Amer. Math. Soc. 120 (1994), 425-434. MR 94d:11089
  • [Lou 6] S. Louboutin. Majorations explicites du résidu au point $1$ des fonctions zêta des corps de nombres. to appear in the Journal of the Mathematical Society of Japan.
  • [Lou 7] S. Louboutin. Determination of all quaternion octic CM-fields with class number $2.$ J. London Math. Soc. (2) 54 (1996), 227-238. CMP 96:17.
  • [Lou-Oka] S. Louboutin and R. Okazaki. Determination of all non-normal quartic CM-fields and of all non-abelian normal octic CM-fields with class number one. Acta Arith. 67 (1994), 47-62.MR 95g:11107
  • [Mar] J. Martinet. Sur l'arithmétique des extensions galoisiennes à groupe de galois diédral d'ordre $2p.$ Ann. Inst. Fourier, Grenoble 19 (1969), 1-80.MR 41:6820
  • [MW] H.L. Montgomery and P.J. Weinberger. Notes on small class number. Acta Arith. 24 (1974), 529-542. MR 50:9841
  • [Odl] A.M. Odlyzko. Some analytic estimates of class numbers and discriminants. Inventiones Math. 29 (1975), 279-286.MR 51:12788
  • [Ol] M. Olivier. Table des corps cubiques réels de discriminant inférieur à $50000,$ avec base d'entiers, nombre de classes, régulateur et unités fondamentales. (January 1992), private communication.
  • [Set] B. Setzer. The determination of all imaginary quartic abelian number fields with class number $1.$ Math. Comp. 35 (1980), 1383-1386. MR 81k:12005
  • [Sta 1] H.M. Stark. A complete determination of the complex quadratic fields of class-number one. Michigan Math. J. 14 (1967), 1-27. MR 36:5102
  • [Sta 2] H.M. Stark. On complex quadratic fields with class-number two. Math. Comp. 29 (1975), 289-302. MR 51:5548
  • [Sta 3] H.M. Stark. Some effective cases of the Brauer-Siegel theorem. Invent. Math. 23 (1974), 135-152. MR 49:7218
  • [Sun] J.S. Sunley. Class numbers of totally imaginary quadratic extensions of totally real fields. Trans. Amer. Math. Soc. 175 (1973), 209-232.MR 47:184
  • [Wa] L.C. Washington. Introduction to Cyclotomic Fields. Springer-Verlag, Grad.Texts Math. 83, 1982. MR 85g:11001
  • [Yam] K. Yamamura. The determination of the imaginary abelian number fields with class-number one. Math. Comp. 62 (1994), 899-921.MR 94g:11096

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Additional Information

Stéphane Louboutin
Affiliation: Université de Caen, UFR Sciences, Département de Mathématiques, Esplanade de la paix, 14032 Caen Cedex, France
Email: loubouti@math.unicaen.fr

Ryotaro Okazaki
Affiliation: Doshisha University, Department of Mathematics, Tanabe, Kyoto, 610-03, Japan
Email: rokazaki@doshisha.ac.jp

Michel Olivier
Affiliation: Laboratoire A2X, UMR 99 36, Université Bordeaux I, 351 Cours de la Libération, 33405 Talence Cedex, France
Email: olivier@math.u-bordeaux.fr

DOI: https://doi.org/10.1090/S0002-9947-97-01768-6
Keywords: CM-field, dihedral field, relative class number
Received by editor(s): July 16, 1995
Received by editor(s) in revised form: March 21, 1996
Article copyright: © Copyright 1997 American Mathematical Society

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