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CM-fields with relative class number one


Authors: Geon-No Lee and Soun-Hi Kwon
Journal: Math. Comp. 75 (2006), 997-1013
MSC (2000): Primary 11R29, 11R42
DOI: https://doi.org/10.1090/S0025-5718-05-01811-9
Published electronically: November 29, 2005
MathSciNet review: 2197004
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Abstract: We will show that the normal CM-fields with relative class number one are of degrees $ \leq 216$. Moreover, if we assume the Generalized Riemann Hypothesis, then the normal CM-fields with relative class number one are of degrees $ \leq 96$, and the CM-fields with class number one are of degrees $ \leq 104$. By many authors all normal CM-fields of degrees $ \leq 96$ with class number one are known except for the possible fields of degree $ 64$ or $ 96$. Consequently the class number one problem for normal CM-fields is solved under the Generalized Riemann Hypothesis except for these two cases.


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  • [B] S. Bessassi, Bounds for the degrees of CM-fields of class number one, Acta Arith., 106.3(2003), 213-245. MR 1957106 (2003m:11183)
  • [CK1] K.-Y. Chang and S.-H. Kwon, Class numbers of imaginary abelian number fields, Proc. Amer. Math. Soc.,128(2000), 2517-2528. MR 1707511 (2000m:11108)
  • [CK2] K.-Y. Chang and S.-H. Kwon, The non-abelian normal CM-fields of degree $ 36$ with class number one, Acta Arith., 101(2002), 53-61. MR 1879845 (2003e:11119)
  • [CK3] K.-Y. Chang and S.-H. Kwon, The class number one problem for some non-abelian normal CM-fields of degree $ 48$, Math. Comp., 72(2003), 1003-1017. MR 1954981 (2003m:11185)
  • [H] J. Hoffstein, Some analytic bounds for zeta functions and class numbers, Invent. Math., 55(1979), 37-47. MR 0553994 (80k:12019)
  • [HS] M. Hall, Jr. and J. K. Senior, The groups of order $ 2^n $ ($ n \leq 6 $), Macmillan, NY, 1964. MR 0168631 (29:5889)
  • [JNO] R. James, M. F. Newman, and E. A. O'Brien, The groups of order $ 128 $, J. Algebra, 129(1990), 136-158. MR 1037398 (90j:20050)
  • [Lef] Y. Lefeuvre, Corps di$ \acute{e}$draux $ \grave{a}$ multiplication complexe principaux, Ann. Inst. Fourier (Grenoble), 50(2000), 67-103. MR 1762338 (2001g:11166)
  • [LLO] F. Lemmermeyer, S. Louboutin, and R. Okazaki, The class number one problem for some non-abelian normal CM-fields of degree $ 24$, J. Th$ \acute{e}$or. Nombres Bordeaux, 11(1999), 387-406. MR 1745886 (2001j:11104)
  • [LO1] 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., 67.1(1994), 47-92. MR 1292520 (95g:11107)
  • [LO2] S. Louboutin and R. Okazaki, The class number one problem for some non-abelian normal CM-fields of $ 2$-power degrees, Proceedings London Math. Soc., Vol. 76, Part 3(1998), 523-548. MR 1616805 (99c:11138)
  • [LOO] S. Louboutin, R. Okazaki, and M. Olivier, The class number one problem for some non-abelian normal CM-fields, Trans. Amer. Math. Soc., 349(1997), 3657-3678. MR 1390044 (97k:11149)
  • [Lou1] S. Louboutin, The class number one problem for the non-abelian normal CM-fields of degree $ 16$, Acta Arith., 82.2(1997), 173-196. MR 1477509 (98j:11097)
  • [Lou2] S. Louboutin, Explicit bounds for residues of Dedekind zeta functions, values of L-functions at $ s = 1$ and relative class numbers, J. Number Theory, 85(2000), 263-282. MR 1802716 (2002i:11111)
  • [Lou3] S. Louboutin, Explicit lower bounds for residues at $ s=1$ of Dedekind zeta functions and relative class numbers of CM-fields, Trans. Amer. Math. Soc., 355(2003), 3079-3098. MR 1974676 (2004f:11134)
  • [LPCK] S. Louboutin, Y.-H. Park, K.-Y. Chang, and S.-H. Kwon, The class number one problem for the non-abelian normal CM-fields of degree $ 2pq$, Preprint.
  • [M] G. A. Miller, Determination of all the groups of order $ 64 $, Amer. J. Math., 52(1930), 617-634. MR 1507920
  • [O1] A. M. Odlyzko, Some analytic extimates of class numbers and discriminants, Invent. Math., 29(1975), 275-286. MR 0376613 (51:12788)
  • [O2] A. M. Odlyzko, Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: A Survey of recent results, Sém. Théor. Nombres Bordeaux 2(1990), 119-141. MR 1061762 (91i:11154)
  • [Ob] E. A. O'Brien, The groups of order dividing $ 256$, Bull. Austral. Math. Soc., 39(1989), 159-160.
  • [P] Y.-H. Park, The class number one problem for the non-abelian normal CM-fields of degree $ 24$ and $ 40$, Acta Arith., 101.1(2002), 63-80. MR 1879844 (2002k:11200)
  • [Poi1] G. Poitou, Minorations de discriminants, Sém. Bourbaki, 28e année (1975/76), 136-153. MR 0435033 (55:7995)
  • [Poi2] G. Poitou, Sur les petits discriminants, Sém. Delange-Pisot-Poitou, 18e année (1976/77), 6-01 - 6-17. MR 0551335 (81i:12007)
  • [PK] S.-M. Park and S.-H. Kwon, Class number one problem for normal CM-fields, Preprint.
  • [PYK] S.-M. Park, H.-S. Yang, and S.-H. Kwon, Class number one problem for the normal CM-fields of degree $ 32$, Preprint.
  • [S] H.M. Stark, Some effective cases of the Brauer-Siegel theorem, Invent. Math., 23(1974), 135-152. MR 0342472 (49:7218)
  • [TW] A. D. Thomas and G. V. Wood, Group Tables, Shiva Mathematics series 2, 1980. MR 0572793 (81d:20002)
  • [W] L. C. Washington, Introduction to cyclotomic fields, 2nd ed. (1997), Grad. Texts in Math. 83, Springer-Verlag. MR 1421575 (97h:11130)
  • [Wo] http://mathworld.wolfram.com/Finitegoup.html.
  • [Y] K. Yamamura, The determination of the imaginary abelian number fields with class number one, Math. Comp., 62(1994), 899-921. MR 1218347 (94g:11096)

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Additional Information

Geon-No Lee
Affiliation: Department of Mathematics Education, Korea University, 136-701, Seoul, Korea
Email: thisknow@korea.ac.kr

Soun-Hi Kwon
Affiliation: Department of Mathematics Education, Korea University, 136-701, Seoul, Korea
Email: sounhikwon@korea.ac.kr

DOI: https://doi.org/10.1090/S0025-5718-05-01811-9
Keywords: CM-fields, class numbers, relative class numbers, Dedekind zeta functions
Received by editor(s): January 19, 2005
Received by editor(s) in revised form: February 27, 2005
Published electronically: November 29, 2005
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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