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ANTI-CYCLOTOMIC EXTENSION AND HILBERT CLASS FIELD

  • Oh, Jangheon (Department of Applied Mathematics Sejong University)
  • 발행 : 2012.02.15

초록

In this paper, we show how to construct the first layer $k^{\alpha}_{1}$ of anti-cyclotomic ${\mathbb{{Z}}}_{3}$-extension of imaginary quadratic fields $k(=\;{\mathbb{{Q}}}(\sqrt{-d}))$ when the Sylow subgroup of class group of k is 3-elementary, and give an example. This example is different from the one we obtained before in the sense that when we write $k^{\alpha}_{1}\;=\;k({\eta}),{\eta}$ is obtained from non-units of ${\mathbb{{Q}}}({\sqrt{3d}})$.

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참고문헌

  1. H. Cohen, Advanced Topics in Computational Number Theory, Springer, 1999.
  2. R. Greenberg, On the Iwasawa invariants of totally real number fields, American Journal of Math. 98 (1976), no. 1, 263-284. https://doi.org/10.2307/2373625
  3. J. Minardi, Iwasawa modules for $\mathbb{Z}^{d}_{p}$-extensions of algebraic number fields, Ph.D dissertation, University of Washington, 1986.
  4. J. Oh, On the first layer of anti-cyclotomic $\mathbb{Z}_{p}$-extension over imaginary quadratic fields, Proc. Japan Acad. Ser. A Math. Sci. 83 (2007), no. 3, 19-20. https://doi.org/10.3792/pjaa.83.19
  5. J. Oh, A note on the first layers of $\mathbb{Z}_{p}$-extensions, Commun. Korean Math. Soc. 24 (2009), no. 3, 1-4. https://doi.org/10.4134/CKMS.2009.24.1.001
  6. J. Oh, Construction of 3-Hilbert class field of certain imaginary quadratic fields, Proc. Japan Aca. Ser. A Math. Sci. 86 (2010), no. 1, 18-19 . https://doi.org/10.3792/pjaa.86.18

피인용 문헌

  1. CONSTRUCTION OF THE FIRST LAYER OF ANTI-CYCLOTOMIC EXTENSION vol.21, pp.3, 2013, https://doi.org/10.11568/kjm.2013.21.3.265
  2. ON THE ANTICYCLOTOMIC ℤp-EXTENSION OF AN IMAGINARY QUADRATIC FIELD vol.23, pp.3, 2015, https://doi.org/10.11568/kjm.2015.23.3.323