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The Ramanujan Journal

, Volume 39, Issue 1, pp 49–59 | Cite as

A construction of nonabelian simple étale fundamental groups II

  • Kwang-Seob Kim
Article
  • 119 Downloads

Abstract

In the previous article (Ramanujan J, 35(1):111–120, 2014) we showed that there is a real quadratic field \(K\) such that \(\mathrm {Gal}(K^f_\mathrm{{ur}}/K)\) is isomorphic to \(A_5\) under the assumption of the generalized Riemann hypothesis (GRH). This is a continuation of Kim (Ramanujan J, 35(1):111–120, 2014). In this article we will find more number fields \(K\) such that \(\mathrm {Gal}(K^f_\mathrm{{ur}}/K)\) is nonabelian simple under the assumption of the GRH.

Keywords

Nonabelian étale fundamental groups Nonabelian simple unramified extensions of number fields Class number one problems 

Mathematics Subject Classification

Primary 11R37 Secondary 11F80 11R29 

References

  1. 1.
    Basmaji, J., Kiming, I.: A table of \(A_5\) fields, On Artin’s conjecture for odd \(2\)-dimensional representations. Lecture Notes in Mathematics, vol. 1585, pp. 37–46, 122–141. Springer, Berlin (1994)Google Scholar
  2. 2.
    Cornell, G.: Abhyankar’s lemma and the class group. Number theory, Carbondale. In: Proceedings of Southern Illinois Conference, Southern Illinois University, Carbondale, Ill, 1979. Lecture Notes in Mathematics, vol. 751, pp. 82–88, Springer, Berlin (1979)Google Scholar
  3. 3.
    Chen, D.: Products of simple group involving the alternating \(A_8\). Math. Aeterna 3(5), 329–336 (2013)MATHMathSciNetGoogle Scholar
  4. 4.
    Fu, L.: Etale cohomology theory. Nankai Tracts in Mathematics, vol. 13. World Scientific Publishing Co. Pte. Ltd., Hackensack (2011)Google Scholar
  5. 5.
    Kim, K.: A construction of nonabelian simple étale fundamental groups. Ramanujan J. 35(1), 111–120 (2014)Google Scholar
  6. 6.
    Martinet, J.: Petits discriminants des corps de nombres. Number Theory Days, 1980 (Exeter 1980). London Mathematics Society Lecture Note Series, pp. 151–193. Cambridge University Press, Cambridge, New York (1982)Google Scholar
  7. 7.
    Neukirch, J.: Algebraic number theory. Grundlehren der Mathematischen Wissenschaften. Springer, Berlin (1999)Google Scholar
  8. 8.
    Odlyzko, A.M.: Lower bounds for discriminants of number fields. Acta Arith. 29(3), 275–297 (1976)MATHMathSciNetGoogle Scholar
  9. 9.
    Odlyzko, A.M.: Lower bounds for discriminants of number fields. II. Tohoku Math. J. 29(2), 209–216 (1977)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Odlyzko, A.M.: Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results. Sém. Théor. Nombres Bordeaux, Série 2, 1, 119–141 (1990).Google Scholar
  11. 11.
    Serre, J.P.: Minorations de discriminants. note of October 1975, published on pp. 240–243 in, vol. 3 of Jean-Pierre Serre, Collected Papers, Springer, Berlin (1986)Google Scholar
  12. 12.
    Wilson, R.A.: The finite simple groups. Graduate Texts in Mathematics, vol. 251. Springer, London (2009)Google Scholar
  13. 13.
    Yamamura, K.: Maximal unramified extensions of imaginary quadratic number fields of small conductors. J. Théor. Nombres Bordeaux 9(2), 405–448 (1997)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Yamamura, K.: Maximal unramified extensions of imaginary quadratic number fields of small conductors. II. J. Théor. Nombres Bordeaux 13(2), 633–649 (2001)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.School of MathematicsKorea Institute for Advanced Study (KIAS)SeoulRepublic of Korea

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