The imaginary abelian number fields with class numbers equal to their genus class numbers

Ku-Young Chang; Soun-Hi Kwon

Journal de théorie des nombres de Bordeaux (2000)

  • Volume: 12, Issue: 2, page 349-365
  • ISSN: 1246-7405

Abstract

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We know that there exist only finitely many imaginary abelian number fields with class numbers equal to their genus class numbers. Such non-quadratic cyclic number fields are completely determined in [Lou2,4] and [CK]. In this paper we determine all non-cyclic abelian number fields with class numbers equal to their genus class numbers, thus the one class in each genus problem is solved, except for the imaginary quadratic number fields.

How to cite

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Chang, Ku-Young, and Kwon, Soun-Hi. "The imaginary abelian number fields with class numbers equal to their genus class numbers." Journal de théorie des nombres de Bordeaux 12.2 (2000): 349-365. <http://eudml.org/doc/248484>.

@article{Chang2000,
abstract = {We know that there exist only finitely many imaginary abelian number fields with class numbers equal to their genus class numbers. Such non-quadratic cyclic number fields are completely determined in [Lou2,4] and [CK]. In this paper we determine all non-cyclic abelian number fields with class numbers equal to their genus class numbers, thus the one class in each genus problem is solved, except for the imaginary quadratic number fields.},
author = {Chang, Ku-Young, Kwon, Soun-Hi},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {genus class number; CM fields; abelian fields},
language = {eng},
number = {2},
pages = {349-365},
publisher = {Université Bordeaux I},
title = {The imaginary abelian number fields with class numbers equal to their genus class numbers},
url = {http://eudml.org/doc/248484},
volume = {12},
year = {2000},
}

TY - JOUR
AU - Chang, Ku-Young
AU - Kwon, Soun-Hi
TI - The imaginary abelian number fields with class numbers equal to their genus class numbers
JO - Journal de théorie des nombres de Bordeaux
PY - 2000
PB - Université Bordeaux I
VL - 12
IS - 2
SP - 349
EP - 365
AB - We know that there exist only finitely many imaginary abelian number fields with class numbers equal to their genus class numbers. Such non-quadratic cyclic number fields are completely determined in [Lou2,4] and [CK]. In this paper we determine all non-cyclic abelian number fields with class numbers equal to their genus class numbers, thus the one class in each genus problem is solved, except for the imaginary quadratic number fields.
LA - eng
KW - genus class number; CM fields; abelian fields
UR - http://eudml.org/doc/248484
ER -

References

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  1. [A] S. Arno, The imaginary quadratic fields of class number 4. Acta Arith.60 (1992), 321-334. Zbl0760.11033MR1159349
  2. [CK] K.-Y. Chang, S.-H. Kwon, On the imaginary cyclic number fields. J. Number Theory73 (1998), 318-338. Zbl0920.11077MR1657972
  3. [G] M.-N. Gras, Classes et unités des extensions cycliques réelles de degré 4 de Q. Ann. Inst. Fourier (Grenoble) 29 (1979), 107-124. Zbl0387.12001MR526779
  4. [H] H. Hasse, Über die Klassenzahl abelscher Zahlkörper. Academie-Verlag, Berlin, 1952. Reprinted with an introduction by J. Martinet: Springer-Verlag, Berlin (1985). Zbl0046.26003MR842666
  5. [HY1] M. Hirabayashi, K. Yoshino, Remarks on unit indices of imaginary abelian number fields. Manuscripta Math.60 (1988), 423-436. Zbl0654.12002MR933473
  6. [HY2] M. Hirabayashi, K. Yoshino, Remarks on unit indices of imaginary abelian number fields II. Mamuscripta Math.64 (1989), 235-251. Zbl0703.11054MR998489
  7. [HY3] M. Hirabayashi, K. Yoshino, Unit indices of imaginary abelian number fields of type (2,2,2). J. Number Theory34 (1990), 346-361. Zbl0705.11065MR1049510
  8. [K] T. Kubota, Über den bizyklischen biquadratischen Zahlkörper. Nagoya Math. J.10 (1956), 65-85. Zbl0074.03001MR83009
  9. [KT] M. Daberkow, C. Fieker, J. Klüners, M. Pohst, K. Roegner, K. Wildanger, KANT V4J. Symbolic Comp.24 (1997), 267-283. Zbl0886.11070MR1484479
  10. [L] F. Van Der Linden, Class number computations of real abelian number fields. Math. Comp.39160 (1982), 693-707. Zbl0505.12010MR669662
  11. [Lou1] S. Louboutin, A finiteness theorem for imaginary abelian number fields. Manuscripta Math.91 (1996), 343-352. Zbl0869.11089MR1416716
  12. [Lou2] S. Louboutin, The nonquadratic imaginary cyclic fields of 2-power degrees with class numbers equal to their genus class numbers. Proc. Amer. Math. Soc.27 (1999), 355-361. Zbl0919.11071MR1468198
  13. [Lou3] S. Louboutin, Minoration au point 1 des fonctions L et determination des sextiques abéliens totalement imaginaires principaux. Acta Arith.62 (1992), 535-540. Zbl0761.11041MR1183984
  14. [Lou4] S. Louboutin, The imaginary cyclic sextic fields with class numbers equal to their genus class numbers. Colloquium Math.75 (1998), 205-212. Zbl0885.11057MR1490690
  15. [Lou5] S. Louboutin, Minorations (sous l'hypothese de Riemann généralisée) des nombres de classes des corps quadratiques imaginaires. Application, C. R. Acad. Sci. Paris Sér. I Math.310 (1990), 795-800. Zbl0703.11048MR1058499
  16. [M] I. Miyada, On imaginary abelian number fields of type (2,2,..., 2) with one class in each genus. Manuscripta Math.88 (1995), 535-540. Zbl0851.11061MR1362937
  17. [Ma] J.M. Masley, Class numbers of real cyclic number fields with small conductor. Compositio Math.37 (1978), 297-319. Zbl0428.12003MR511747
  18. [Mä] S. Mäki, The determination of units in real cyclic sextic fields. Lecture notes in Math. 797, Springer-Verlag, Berlin and New York (1980). Zbl0423.12006MR584794
  19. [P] C. Batut, D. Bernardi, H. Cohen, M. Olivier, PARI-GP, version 1. 38. 
  20. [S1] H.M. Stark, A complete determination of the complex quadratic fields of class number one. Michigan Math. J.14 (1967), 1-27. Zbl0148.27802MR222050
  21. [S2] H.M. Stark, On complex quadratic fields with class number two. Math. Comp.29 (1975), 289-302. Zbl0321.12009MR369313
  22. [W] L.C. Washington, Introduction to cyclotomic fields. GTM 83, 2nd Ed., Springer-Verlag (1996). Zbl0966.11047MR1421575
  23. [We] P.J. Weinberger, Exponents of the class groups of complex quadratic fields. Acta Arith.22 (1973), 117-124. Zbl0217.04202MR313221
  24. [Y] K. Yamamura, The determination of imaginary abelian number fields with class number one. Math. Comp.62 (1994), 899-921. Zbl0798.11046MR1218347
  25. [YH1l K. Yoshino, M. Hirabayashi, On the relative class number of the imaginary abelian number field I. Memoirs of the College of Liberal Arts, Kanazawa Medical University, Vol. 9 (1981). Zbl0667.12003
  26. [YH2] K. Yoshino, M. Hirabayashi, On the relative class number of the imaginary abelian number field II. Memoirs of the College of Liberal Arts, Kanazawa Medical University, Vol. 10 (1982) . Zbl0667.12003

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