Skip to main content

On the construction of normal wildly ramified extensions over Q p , (p ≠ 2)

Abstract

Let p be an odd prime number, and let Q p be the field of rational p-adic numbers.The aim of this work is the determination of the standard form of an Eisenstein polynomial defining a normal wildly ramified extension of Q p . We prove first the equivalence between normality and cyclicity, give some essential normality conditions for the general case (degree p n), then we solve the problem completely for the case (degree p 2) also, we obtain that the normality depends on seven congruences modulo p m between the coefficients of the considered polynomial with just m = 2 or 3. Note that the case (degree p) was solved by Öystein Ore (see Math. Annalen 102 (1930), 283–304). Also examples are given.

This is a preview of subscription content, access via your institution.

References

  1. E. Artin, Algebraic Numbers and Algebraic Functions, Lecture notes Princeton University and New York University 1950–1951, Gordon and Breach 1967.

  2. J. Cassels and A. Fröhlich, Algebraic Numbers Theory, Academic press Inc., U.S. (1967) London–New York.

  3. Krasner M.: Sur la primitivité des corps p-adic. Mathematica Cluj 13, 72–191 (1937)

    MATH  Google Scholar 

  4. Ö. Ore, Abriss einer arithmetischen Theorie der Galoisschen Körper, Math. Annalen Teil 1. 100 (1928) 650–673, Teil 2. 102 (1930) 283–304.

  5. Serre J.P.: Corps Locaux. Publications de l’Institut de Mathématique de l’Université de Nancago, Hermann (1962)

    MATH  Google Scholar 

  6. N. Kobliz, P-adic Numbers, P-adic Analysis and Zeta functions. Springer- Verlag, Berlin, New York, London, Tokyo, Second edition 1984.

  7. Lbekkouri A.: On the construction of normal wildly ramified extensions over Q 2. Arch. Math. 93, 234–243 (2009)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Akram Lbekkouri.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Lbekkouri, A. On the construction of normal wildly ramified extensions over Q p , (p ≠ 2). Arch. Math. 93, 331 (2009). https://doi.org/10.1007/s00013-009-0026-3

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1007/s00013-009-0026-3

Keywords

  • Galois Group
  • Sylow Subgroup
  • Algebraic Closure
  • Minimal Polynomial
  • Newton Polygon