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A simple construction of genus fields of abelian number fields


Author: Xian Ke Zhang
Journal: Proc. Amer. Math. Soc. 94 (1985), 393-395
MSC: Primary 11R20
MathSciNet review: 787879
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Abstract: Simple elementary construction of the genus field $ {K^ * }$ (= maximal abelian subfield of the Hilbert class field) of any abelian number field $ K$ is given without using class field theory. When $ K$ is of type $ (l, \ldots ,l)$ with $ l$ prime, the construction is more explicit. These results contain some former results and show that the main result in [8] has mistakes.


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

  • [1] Heinrich W. Leopoldt, Zur Geschlechtertheorie in abelschen Zahlkörpern, Math. Nachr. 9 (1953), 351–362 (German). MR 0056032
  • [2] Makoto Ishida, The genus fields of algebraic number fields, Lecture Notes in Mathematics, Vol. 555, Springer-Verlag, Berlin-New York, 1976. MR 0435028
  • [3] C. S. Herz, Construction of class fields, Lecture Notes in Math., Vol. 21, Springer-Verlag, Berlin and New York, 1966.
  • [4] Daniel A. Marcus, Number fields, Springer-Verlag, New York-Heidelberg, 1977. Universitext. MR 0457396
  • [5] Zhang Xianke, On number fields of type $ (2, \ldots ,2)$, J. China Univ. Sci. Tech. 12 (1982), 29-41.
  • [6] -, On number fields of type $ (l, \ldots ,l)$, Sci. Sinica Ser. A 1 (1984), 31-38.
  • [7] Yoshihiro Kubokawa, The genus field for composite of quadratic fields, J. Saitama Univ. Fac. Ed. Math. Natur. Sci. 26 (1977), 1–3 (1978). MR 546256
  • [8] M. Bhaskaran, Construction of genus field and some applications, J. Number Theory 11 (1979), no. 4, 488–497. MR 544896, 10.1016/0022-314X(79)90028-3

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1985-0787879-0
Keywords: Number field, genus field, abelian extension
Article copyright: © Copyright 1985 American Mathematical Society