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A Polynomial Time Nilpotence Test for Galois Groups and Related Results

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Mathematical Foundations of Computer Science 2006 (MFCS 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4162))

Abstract

We give a deterministic polynomial-time algorithm to check whether the Galois group Gal(f) of an input polynomial f(X) ∈ ℚ[X] is nilpotent: the running time is polynomial in size(f). Also, we generalize the Landau-Miller solvability test to an algorithm that tests if Gal(f) is in Γ d : this algorithm runs in time polynomial in size(f) and n d and, moreover, if Gal(f) ∈ Γ d it computes all the prime factors of # Gal(f).

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References

  1. Babai, L., Cameron, P.J., Pálfy, P.P.: On the order of primitive groups with restricted nonabelian composition factors. Journal of Algebra 79, 161–168 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  2. Fernandez-Ferreiros, P., Gomez-Molleda, M.A.: Deciding the nilpotency of the galois group by computing elements in the centre. Mathematics of Computation 73(248) (2003)

    Google Scholar 

  3. Landau, S.: Polynomial time algorithms for galois groups. In: Fitch, J. (ed.) EUROSAM 1984 and ISSAC 1984. LNCS, vol. 174, pp. 225–236. Springer, Heidelberg (1984)

    Chapter  Google Scholar 

  4. Landau, S., Miller, G.L.: Solvability by radicals is in polynomial time. Journal of Computer and System Sciences 30, 179–208 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  5. Lang, S.: Algebra, 3rd edn. Addison-Wesley Publishing Company, Inc., Reading (1999)

    Google Scholar 

  6. Lenstra Jr., H.W.: Algorithms in algebraic number theory. Bulletin of the American Mathematical Society 26(2), 211–244 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  7. Luks, E.M.: Isomorphism of graphs of bounded valence can be tested in polynomial time. Journal of Computer and System Sciences 25(1), 42–65 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  8. Luks, E.M.: Permutation groups and polynomial time computations. DIMACS Series in Discrete Mathematics and Theoretical Computer Science 11, 139–175 (1993)

    MathSciNet  Google Scholar 

  9. Wielandt, H.: Finite Permutation Groups. Academic Press, New York (1964)

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Institute of Mathematical Sciences, C.I.T Campus, Chennai, 600 113, India

    V. Arvind

  2. Department of Computer Science and Engineering, Indian Institute of Technology, Kanpur, Kanpur, 208016, UP, India

    Piyush P Kurur

Authors
  1. V. Arvind
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  2. Piyush P Kurur
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Editor information

Editors and Affiliations

  1. Department of Computer Science, Comenius University, Mlynská dolina, 84248, Bratislava, Slovakia

    Rastislav Královič

  2. Institute of Informatics, University of Warsaw, Poland

    Paweł Urzyczyn

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Arvind, V., Kurur, P.P. (2006). A Polynomial Time Nilpotence Test for Galois Groups and Related Results. In: Královič, R., Urzyczyn, P. (eds) Mathematical Foundations of Computer Science 2006. MFCS 2006. Lecture Notes in Computer Science, vol 4162. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11821069_12

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  • DOI: https://doi.org/10.1007/11821069_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-37791-7

  • Online ISBN: 978-3-540-37793-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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