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On the computation of class numbers of real abelian fields


Author: Tuomas Hakkarainen
Journal: Math. Comp. 78 (2009), 555-573
MSC (2000): Primary 11R29, 11Y40; Secondary 11R20, 11R27
DOI: https://doi.org/10.1090/S0025-5718-08-02169-8
Published electronically: September 4, 2008
MathSciNet review: 2448721
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we give a procedure to search for prime divisors of class numbers of real abelian fields and present a table of odd primes $ <10000$ not dividing the degree that divide the class numbers of fields of conductor $ \leq 2000$. Cohen-Lenstra heuristics allow us to conjecture that no larger prime divisors should exist. Previous computations have been largely limited to prime power conductors.


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  • 1. M. Aoki, Notes on the structure of the ideal class groups of abelian number fields, Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), no. 5, pp. 69-74. MR 2143545 (2006a:11142)
  • 2. J. Buhler, C. Pomerance, L. Robertson, Heuristics for class numbers of prime-power real cyclotomic fields, High primes and misdemeanours: lectures in honour of the 60th birthday of Hugh Cowie Williams, Fields Inst. Commun. 41, Amer. Math. Soc., Providence, RI (2004), pp. 149-157. MR 2073643 (2005e:11143)
  • 3. H. Cohen, H. W. Lenstra, Heuristics on class groups of number fields, Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983), Lecture Notes in Math. 1068, Springer, Berlin (1984), pp. 33-62. MR 756082 (85j:11144)
  • 4. C.-E. Fröberg, On the prime zeta function, BIT 8 (1968), pp. 187-202. MR 0236123 (38:4421)
  • 5. G. Gras and M.-N. Gras, Calcul du nombre de classes et des unités des extensions abéliennes réelles de $ \mathbf{Q}$, Bull. Sci. Math. (2) 101 (1977), no. 2, pp. 97-129. MR 0480423 (58:586)
  • 6. M.-N. Gras, Méthodes et algorithmes pour le calcul numérique du nombre de classes et des unités des extensions cubiques cycliques de $ \mathbf{Q}$, J. Reine Angew. Math. 277 (1975), pp. 89-116. MR 0389845 (52:10675)
  • 7. M.-N. Gras, Table numérique du nombre de classes et des unités dans les extensions cycliques réelles de degre $ 4$ de $ \mathbf{Q}$, Publ. Math. Fac. Sci. Besançon 1977/78, Fasc. 2 (1978), 52 pp.
  • 8. T. Hakkarainen, On the computation of the class numbers of real abelian fields, TUCS Dissertations no. 87, Turku Centre for Computer Science (2007), 81 pp. Available at http://www.tucs.fi/
  • 9. S. Jeannin, Tables des nombres de classes et unités des corps quintiques cycliques de conducteur $ f\leq 10000$, Publ. Math. Fac. Sci. Besançon 1994/95-1995/96 (1997), 40 pp. MR 1449427 (98b:11129)
  • 10. S. Kobayashi, Divisibilité du nombre de classes des corps abéliens réels, J. Reine Angew. Math. 320 (1980), pp. 142-149. MR 592150 (82f:12009)
  • 11. Y. Koyama and K. Yoshino, Prime divisors of real class number of $ p^r$th cyclotomic field and characteristic polynomials attached to them, Preprint (2003), 23 pp.
  • 12. H. W. Leopoldt, Über Einheitengruppe und Klassenzahl reeller abelscher Zahlkörper, Abh. Deutsch. Akad. Wiss. Berlin. Kl. Math. Nat. 1953, no. 2 (1954), 48 pp. MR 0067927 (16:799d)
  • 13. H. W. Leopoldt, Über Klassenzahlprimteiler reeller abelscher Zahlkörper als Primteiler verallgemeinerter Bernoullischer Zahlen, Abh. Math. Sem. Univ. Hamburg 23 (1959), pp. 36-47. MR 0103184 (21:1967)
  • 14. F. van der Linden, Class number computations of real abelian number fields, Math. Comp. 39 (1982), pp. 693-707. MR 669662 (84e:12005)
  • 15. S. Mäki, The determination of units in real cyclic sextic fields, Lecture Notes in Math. 797, Springer, Berlin (1980), 198 pp. MR 584794 (82a:12004)
  • 16. T. Metsänkylä, An application of the $ p$-adic class number formula, Manuscripta Math. 93 (1997), pp. 481-498. MR 1465893 (98m:11118)
  • 17. B. Oriat, Groupes des classes d'idéaux des corps quadratiques réels $ \mathbf{Q}(d^{1/2}),\, 1<d<24572$, Publ. Math. Fac. Sci. Besançon 1986/87-1987/88, Fasc. 2 (1988), 65 pp. MR 983124 (90e:11167a)
  • 18. PARI/GP, version 2.2.8, Bordeaux, 2005, http://pari.math.u-bordeaux.fr/
  • 19. S. Perlis and G. Walker, Abelian group algebras of finite order, Trans. Amer. Math. Soc. 68 (1950), pp. 420-426. MR 0034758 (11:638k)
  • 20. R. Schoof, Class numbers of real cyclotomic fields of prime conductor, Math. Comp. 72 (2003), pp. 913-937. MR 1954975 (2004f:11116)
  • 21. W. Schwarz, Über die Klassenzahl abelscher Zahlkörper, Ph.D. Thesis, University of Saarbrücken (1995), 125 pp.
  • 22. L. Washington, Introduction to Cyclotomic Fields, 2nd ed., Springer, New York, 1997. MR 1421575 (97h:11130)
  • 23. C. Wittmann, $ p$-class groups of certain extensions of degree $ p$, Math. Comp. 74 (2005), pp. 937-947. MR 2114656 (2005h:11256)
  • 24. Wolfram Research, Inc., Mathematica, Version 4.1, Champaign, IL (2001).

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Additional Information

Tuomas Hakkarainen
Affiliation: Department of Mathematics & TUCS, Turku Centre for Computer Science, University of Turku, FI-20014 Turku, Finland

DOI: https://doi.org/10.1090/S0025-5718-08-02169-8
Keywords: Class numbers, computation, abelian fields, units
Received by editor(s): April 28, 2006
Published electronically: September 4, 2008
Additional Notes: This work was financially supported by the Turku Centre for Computer Science, TUCS
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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