Illinois Journal of Mathematics

Exact algorithms for $p$-adic fields and epsilon constant conjectures

Werner Bley and Manuel Breuning

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We describe an algorithmic approach to prove or disprove several recent conjectures for epsilon constants of Galois extensions of $p$-adic fields and number fields. For this approach, we must develop various algorithms for computations in Galois extensions of $p$-adic fields which are of independent interest. Our algorithms for $p$-adic fields are based on existing algorithms for number fields and are exact in the sense that we do not need to consider approximations to $p$-adic numbers.

Article information

Illinois J. Math., Volume 52, Number 3 (2008), 773-797.

First available in Project Euclid: 1 October 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11Y40: Algebraic number theory computations
Secondary: 11S23: Integral representations 11S25: Galois cohomology [See also 12Gxx, 16H05]


Bley, Werner; Breuning, Manuel. Exact algorithms for $p$-adic fields and epsilon constant conjectures. Illinois J. Math. 52 (2008), no. 3, 773--797. doi:10.1215/ijm/1254403714.

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  • V. Acciaro and J. Klüners, Computing local Artin maps, and solvability of norm equations, J. Symbolic Comp. 30 (2000), 239–251.
  • H. Anai, M. Noro and K. Yokoyama, Computation of the splitting fields and the Galois groups of polynomials, Algorithms in algebraic geometry and applications (L. González-Vega and T. Recio, eds.), Prog. Math., vol. 143, Birkhäuser, 1996, pp. 29–50.
  • W. Bley, Numerical evidence for a conjectural generalization of Hilbert's Theorem 132, LMS J. Comput. Math. 6 (2003), 68–88 (electronic).
  • W. Bley and R. Boltje, Computation of locally free class groups, Algorithmic number theory (F. Hess, S. Pauli and M. Pohst, eds.), Lecture Notes in Computer Science, vol. 4076, Springer, 2006, pp. 72–86.
  • W. Bley and D. Burns, Equivariant epsilon constants, discriminants and étale cohomology, Proc. Lond. Math. Soc. 87 (2003), 545–590.
  • W. Bley and S. M. J. Wilson, Computations in relative algebraic $K$-groups, Preprint.
  • R. Boltje, A canonical Brauer induction formula, Astérisque 181-182 (1990), 31–59.
  • M. Breuning, On equivariant global epsilon constants for certain dihedral extensions, Math. Comp. 73 (2004), 881–898.
  • M. Breuning, Equivariant local epsilon constants and étale cohomology, J. Lond. Math. Soc. 70 (2004), 289–306.
  • T. Chinburg, Exact sequences and Galois module structure, Ann. of Math. 121 (1985), 351–376.
  • H. Cohen, A course in computational algebraic number theory, Springer Verlag, 1993.
  • H. Cohen, Advanced topics in computational number theory, Springer Verlag, 2000.
  • P. Deligne, Les constantes des équations fonctionnelles des fonctions $L$, Modular functions of one variable, vol. II, Springer, 1973, pp. 501–597.
  • K. Geissler and J. Klüners, Galois group computation for rational polynomials, J. Symbolic Comp. 30 (2000), 653–674.
  • K. Girstmair, An algorithm for the construction of a normal basis, J. Number Theory 78 (1999), 36–45.
  • G. Henniart, Relèvement global d'extensions locales: quelques problèmes de plongement, Math. Ann. 319 (2001), 75–87.
  • D. Holt, Cohomology and group extensions in Magma, Discovering mathematics with Magma (W. Bosma and J. Cannon, eds.), Springer, 2006, pp. 221–241.
  • J. J. Hooper and S. M. J. Wilson, Chinburg's second conjecture for $H_{12}$-extensions of $\mathbb{Q}$, Preprint.
  • N. Koblitz, $p$-adic numbers, $p$-adic analysis, and zeta-functions, 2nd ed., Springer Verlag, 1984.
  • F. Lorenz, Einführung in die Algebra II, Spektrum Akademischer Verlag, Heidelberg, 1997.
  • J. Martinet, Character theory and Artin $L$-functions, Algebraic number fields (A. Fröhlich, ed.), Academic Press, 1977, pp. 1–87.
  • J. Neukirch, Algebraische Zahlentheorie, Springer-Verlag, Berlin, 1992.
  • J. Neukirch, A. Schmidt and K. Wingberg, Cohomology of number fields, Springer Verlag, 2000.
  • S. Pauli and X.-F. Roblot, On the computation of all extensions of a $p$-adic field of a given degree, Math. Comp. 70 (2001), 1641–1659.
  • J.-P. Serre, Linear representations of finite groups, Springer Verlag, 1977.
  • J.-P. Serre, Local fields, Springer Verlag, 1979.
  • R. G. Swan, Algebraic $K$-theory, Lecture Notes in Mathematics, vol. 76, Springer Verlag, 1968.
  • K. Yokoyama, A modular method for computing the Galois groups of polynomials, J. Pure Appl. Algebra 117/118 (1997), 617–636.