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International Algorithmic Number Theory Symposium

ANTS 2006: Algorithmic Number Theory pp 56–71Cite as

An Algorithm for Computing p-Class Groups of Abelian Number Fields

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 4076))

Abstract

For an abelian number field F and an odd prime number p which does not divide the degree [F:ℚ], we propose a new algorithm for computing the p-primary part of the ideal class group of F using Gauss sums and cyclotomic units.

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References

  1. Bach, E., Sorenson, J.: Explicit bounds for primes in residue classes. Math. Comp. 65, 1717–1735 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  2. Cornacchia, P.: The 2-ideal class groups of Q(ζ ℓ). Nagoya Math. J. 162, 1–18 (2001)

    MATH  MathSciNet  Google Scholar 

  3. Greenberg, R.: On p-adic L-functions and cyclotomic fields II. Nagoya Math. J. 67, 139–158 (1977)

    MATH  MathSciNet  Google Scholar 

  4. Kolyvagin, V.: Euler systems, in The Grothendieck Festschrift II. Prog. Math. 87, 435–483 (1990)

    MathSciNet  Google Scholar 

  5. Lang, S.: Cyclotomic Fields I and II. Graduate Texts in Mathematics, vol. 121. Springer, Heidelberg (1990)

    MATH  Google Scholar 

  6. Mazur, B., Wiles, A.: Class fields of abelian extensions of ℚ. Invent. math. 76, 179–330 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  7. Rubin, K.: The main conjecture, Appendix to Lang [5]

    Google Scholar 

  8. Rubin, K.: Kolyvagin’s system of Gauss sums, in Arithmetic Algebraic Geometry. Prog. Math. 89, 309–324 (1991)

    Google Scholar 

  9. Schoof, R.: Minus class groups of the fields of the ℓ-th roots of unity. Math. Comp. 67, 1225–1245 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  10. Schoof, R.: Class numbers of real cyclotomic fields of prime conductor. Math. Comp. 72, 913–937 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  11. Sumida, H.: Computation of Iwasawa invariants of certain real abelian fields. J. Number Theory 105, 235–250 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. Thaine, F.: On the ideal class groups of real abelian number fields. Ann. Math. 128, 1–18 (1988)

    Article  MathSciNet  Google Scholar 

  13. Washington, L.: Introduction to Cyclotomic Fields, 2nd edn. Graduate Texts in Mathematics, vol. 83. Springer, Heidelberg (1997)

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Oh-okayama, Meguro-ku, Tokyo, Japan

    Miho Aoki

  2. Department of Mathematics, College of Industrial Technology, Nihon University, 2-11-1 Shin-ei, Narashino, Chiba, Japan

    Takashi Fukuda

Authors
  1. Miho Aoki
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  2. Takashi Fukuda
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Editor information

Editors and Affiliations

  1. Technische Universität Berlin, Germany

    Florian Hess

  2. Institut für Mathematik, MA 8–1, Technische Universität Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Sebastian Pauli

  3. Institut für Mathematik, Technische Universität Berlin, Fakultät II, MA 8-1, Strasse des 17. Juni 136, 10623, Berlin, Germany

    Michael Pohst

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© 2006 Springer-Verlag Berlin Heidelberg

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Aoki, M., Fukuda, T. (2006). An Algorithm for Computing p-Class Groups of Abelian Number Fields. In: Hess, F., Pauli, S., Pohst, M. (eds) Algorithmic Number Theory. ANTS 2006. Lecture Notes in Computer Science, vol 4076. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11792086_5

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  • DOI: https://doi.org/10.1007/11792086_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-36075-9

  • Online ISBN: 978-3-540-36076-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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