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Upper Bounds on the Complexity of Some Galois Theory Problems

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Algorithms and Computation (ISAAC 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2906))

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Abstract

Assuming the generalized Riemann hypothesis, we prove the following complexity bounds: The order of the Galois group of an arbitrary polynomial f(x) ∈ ℤ[x] can be computed in P# P. Furthermore, the order can be approximated by a randomized polynomial-time algorithm with access to an NP oracle. For polynomials f with solvable Galois group we show that the order can be computed exactly by a randomized polynomial-time algorithm with access to an NP oracle. For all polynomials f with abelian Galois group we show that a generator set for the Galois group can be computed in randomized polynomial time.

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Author information

Authors and Affiliations

  1. Institute of Mathematical Sciences, C.I.T Campus, Chennai, 600113, India

    V. Arvind & 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. School of Science and Technology, Kwansei Gakuin University, Sanda, Japan

    Toshihide Ibaraki

  2. Department of Architecture and Architectural Engineering, Kyoto University, 615-8540, Nishikyo-ku, Kyoto, Japan

    Naoki Katoh

  3. Department of Computer Science and Communication Engineering, Kyushu University, 812-8581, Fukuoka, Japan

    Hirotaka Ono

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© 2003 Springer-Verlag Berlin Heidelberg

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Arvind, V., Kurur, P.P. (2003). Upper Bounds on the Complexity of Some Galois Theory Problems. In: Ibaraki, T., Katoh, N., Ono, H. (eds) Algorithms and Computation. ISAAC 2003. Lecture Notes in Computer Science, vol 2906. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24587-2_73

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  • DOI: https://doi.org/10.1007/978-3-540-24587-2_73

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-20695-8

  • Online ISBN: 978-3-540-24587-2

  • eBook Packages: Springer Book Archive

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