Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 

 

On the computation of all extensions of a $p$-adic field of a given degree


Authors: Sebastian Pauli and Xavier-François Roblot
Journal: Math. Comp. 70 (2001), 1641-1659
MSC (2000): Primary 11S15, 11S05; Secondary 11Y40
Published electronically: March 8, 2001
MathSciNet review: 1836924
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Abstract | References | Similar Articles | Additional Information

Abstract:

Let $\mathbf{k}$ be a $p$-adic field. It is well-known that $\mathbf{k}$ has only finitely many extensions of a given finite degree. Krasner has given formulae for the number of extensions of a given degree and discriminant. Following his work, we present an algorithm for the computation of generating polynomials for all extensions $\mathbf{K}/\mathbf{k}$ of a given degree and discriminant.


References [Enhancements On Off] (What's this?)

  • [Ba99] C. Batut, K. Belabas, D. Bernardi, H. Cohen, M. Olivier. The Computer Algebra System PARI-GP, Université Bordeaux I, 1999, ftp://megrez.math.u-bordeaux.fr/pub/pari/
  • [Ca86] J. W. S. Cassels, Local fields, London Mathematical Society Student Texts, vol. 3, Cambridge University Press, Cambridge, 1986. MR 861410
  • [Da96] M. Daberkow, C. Fieker, J. Klüners, M. Pohst, K. Roegner, M. Schörnig, and K. Wildanger, KANT V4, J. Symbolic Comput. 24 (1997), no. 3-4, 267–283. Computational algebra and number theory (London, 1993). MR 1484479, 10.1006/jsco.1996.0126
  • [Ha69] H. Hasse, Number Theory, Grundlehren der mathematischen Wissenschaften 229, Springer-Verlag, Berlin, 1980. MR 81c:12001
  • [He94] Volker Heiermann, De nouveaux invariants numériques pour les extensions totalement ramifiées de corps locaux, J. Number Theory 59 (1996), no. 1, 159–202 (French, with French summary). MR 1399703, 10.1006/jnth.1996.0092
  • [Kr66] Marc Krasner, Nombre des extensions d’un degré donné d’un corps 𝔭-adique, Les Tendances Géom. en Algèbre et Théorie des Nombres, Editions du Centre National de la Recherche Scientifique, Paris, 1966, pp. 143–169 (French). MR 0225756
  • [Kr79] Marc Krasner, Remarques au sujet d’une note de J.-P. Serre: “Une ‘formule de masse’ pour les extensions totalement ramifiées de degré donné d’un corps local” [C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 22, A1031–A1036; MR 80a:12018]: une démonstration de la formule de M. Serre à partir de mon théorème sur le nombre des extensions séparables d’un corps valué localement compact, qui sont d’un degré et d’une différente donnés, C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 18, A863–A865 (French, with English summary). MR 538991
  • [Or26] Ö. Ore, Bemerkungen zur Theorie der Differente, Math. Zeitschr. 25 (1926), pp. 1-8.
  • [Pa95] P. Panayi, Computation of Leopoldt's p-adic regulator, PhD thesis, University of East Anglia, 1995, http://www.mth.uea.ac.uk/~ h090/.
  • [Se63] Jean-Pierre Serre, Corps locaux, Publications de l’Institut de Mathématique de l’Université de Nancago, VIII, Actualités Sci. Indust., No. 1296. Hermann, Paris, 1962 (French). MR 0150130
  • [Se78] Jean-Pierre Serre, Une “formule de masse” pour les extensions totalement ramifiées de degré donné d’un corps local, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 22, A1031–A1036 (French, with English summary). MR 500361
  • [Sh47] I. Shafarevitch, On 𝑝-extensions, Rec. Math. [Mat. Sbornik] N.S. 20(62) (1947), 351–363 (Russian, with English summary). MR 0020546
  • [Ya95] Masakazu Yamagishi, On the number of Galois 𝑝-extensions of a local field, Proc. Amer. Math. Soc. 123 (1995), no. 8, 2373–2380. MR 1264832, 10.1090/S0002-9939-1995-1264832-0

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Additional Information

Sebastian Pauli
Affiliation: Centre Interuniversitaire en Calcul Mathématique Algébrique, Concordia University, 1455 de Maisonneuve Blvd. W., Montréal, Québec, H3G 1M8, CANADA
Email: pauli@cicma.concordia.ca

Xavier-François Roblot
Affiliation: Institut Girard Desargues, Université Claude Bernard (Lyon 1), 43, boulevard du 11 Novembre 1918, 69622 Villeurbanne cedex, France
Email: roblot@desargues.univ-lyon1.fr

DOI: http://dx.doi.org/10.1090/S0025-5718-01-01306-0
Keywords: $p$-adic fields, wildly ramified extensions, Eisenstein polynomials
Received by editor(s): May 24, 1999
Received by editor(s) in revised form: January 14, 2000
Published electronically: March 8, 2001
Additional Notes: The work of the first author was supported in part by ISM and FCAR/CICMA (Québec).
The work of the second author was supported in part by NSERC (Canada) and FCAR/CICMA (Québec).
We would like to thank David Ford for his careful reading of the original manuscript and for his useful comments.
Article copyright: © Copyright 2001 American Mathematical Society