Advertisement

The Ramanujan Journal

, Volume 47, Issue 1, pp 47–66 | Cite as

Mixed degree number field computations

  • John W. Jones
  • David P. Roberts
Article
  • 16 Downloads

Abstract

We present a method for computing complete lists of number fields in cases where the Galois group, as an abstract group, appears as a Galois group in smaller degree. We apply this method to find the 25 octic fields with Galois group \({{\mathrm{PSL}}}_2(7)\) and smallest absolute discriminant. We carry out a number of related computations, including determining the octic field with Galois group \(2^3{:}{{\mathrm{GL}}}_3(2)\) of smallest absolute discriminant.

Keywords

Number field Galois group Discriminant Ramification 

Mathematics Subject Classification

Primary: 11R21 Secondary: 11Y40 11R32 

References

  1. 1.
    Bhargava, M.: The density of discriminants of quintic rings and fields. Ann. Math. (2) 172(3), 1559–1591 (2010)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bhargava, M., Cojocaru, A.C., Thorne, F.: The number of non-\(S_5\) quintic extensions of bounded discriminant. Preprint. March 2015Google Scholar
  3. 3.
    Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24(3–4), 235–265. Computational algebra and number theory (London, 1993) (1997)Google Scholar
  4. 4.
    Cannon, J.J., Holt, D.F.: The transitive permutation groups of degree 32. Exp. Math. 17(3), 307–314 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cohen, H.: Advanced Topics in Computational Number Theory. Graduate Texts in Mathematics, vol. 193. Springer, New York (2000)CrossRefGoogle Scholar
  6. 6.
    Conway, J.H., Hulpke, A., McKay, J.: On transitive permutation groups. LMS J. Comput. Math. 1, 1–8 (1998). (electronic)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ford, D., Pohst, M.: The totally real \(A_5\) extension of degree \(6\) with minimum discriminant. Exp. Math. 1(3), 231–235 (1992)CrossRefGoogle Scholar
  8. 8.
    Ford, D., Pohst, M., Daberkow, M., Haddad, N.: The \(S_5\) extensions of degree 6 with minimum discriminant. Exp. Math. 7(2), 121–124 (1998)CrossRefGoogle Scholar
  9. 9.
    Iwasawa, K.: On Galois groups of local fields. Trans. Am. Math. Soc. 80, 448–469 (1955)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Jones, J.W., Roberts, D.P.: Sextic number fields with discriminant \((-1)^j2^a3^b\). Number Theory (Ottawa, ON, 1996), CRM Proc. Lecture Notes, vol. 19, pp. 141–172. American Mathematical Society, Providence (1996)Google Scholar
  11. 11.
    Jones, J.W., Roberts, D.P: Timing analysis of targeted Hunter searches. Algorithmic Number Theory (Portland, OR, 1998), pp. 412–413 (1998)Google Scholar
  12. 12.
    Jones, J.W., Roberts, D.P.: Septic fields with discriminant \(\pm 2^a3^b\). Math. Comput. 72(244), 1975–1985 (2003). (electronic)CrossRefGoogle Scholar
  13. 13.
    Jones, J.W., Roberts, D.P.: A database of local fields. J. Symb. Comput. 41(1), 80–97 (2006). http://math.asu.edu/~jj/localfields MathSciNetCrossRefGoogle Scholar
  14. 14.
    Jones, J.W., Roberts, D.P: A database of number fields. LMS J. Comput. Math. 17(1), 595–618 (2014). http://hobbes.la.asu.edu/NFDB MathSciNetCrossRefGoogle Scholar
  15. 15.
    Jones, J.W., Roberts, D.P.: The tame-wild principle for discriminant relations for number fields. Algebra Number Theory 8(3), 609–645 (2014)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Jones, J.W., Roberts, D.P.: Artin \(L\)-functions of small conductor. Res. Number Theory 3, Art. 16, 33 (2017)Google Scholar
  17. 17.
    Klüners, J.: A counterexample to Malle’s conjecture on the asymptotics of discriminants. C. R. Math. Acad. Sci. Paris 340(6), 411–414 (2005)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Klüners, J., Malle, G.: A database for field extensions of the rationals. LMS J. Comput. Math. 4, 182–196 (2001). (electronic)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Malle, G.: On the distribution of Galois groups. II. Exp. Math. 13(2), 129–135 (2004)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Schwarz, A., Pohst, M., Diaz y Diaz, F.: A table of quintic number fields. Math. Comput. 63(207), 361–376 (1994)MathSciNetCrossRefGoogle Scholar
  21. 21.
    The LMFDB Collaboration: The L-functions and modular forms database (2018). http://www.lmfdb.org (Online). Accessed Feb 2018
  22. 22.
    The PARI Group: Bordeaux, PARI/GP version 2.7.5, 2015. http://pari.math.u-bordeaux.fr/
  23. 23.
    Trinks, W.: Ein Beispiel eines Zahlkörpers mit der Galoisgruppe \(PSL(3,2)\) über \(\mathbf{Q}\). Univ. Karlsruhe, Karlsruhe (1968)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical and Statistical SciencesArizona State UniversityTempeUSA
  2. 2.Division of Science and MathematicsUniversity of Minnesota-MorrisMorrisUSA

Personalised recommendations