manuscripta mathematica

, Volume 55, Issue 1, pp 39–67 | Cite as

Immediate and purely wild extensions of valued fields

  • Franz -Viktor Kuhlmann
  • Matthias Pank
  • Peter Roquette
Article

Abstract

Kaplansky's hypothesis A concerning valued fields is put into a Galois theoretic setting. Accordingly, Kaplansky's theorem on maximal immediate extensions can be deduced from the Schur-Zassenhaus theorem about conjugacy of complements in profinite groups. Some generalization of Kaplansky's theory is given, concerning maximal purely wild extensions.

Keywords

Number Theory Algebraic Geometry Topological Group Profinite Group Theoretic Setting 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Ax, J. - Kochen, S.: Diophantine problems over local fields, Amer. Journ. Math. 87, 605–648 (1965)Google Scholar
  2. [2]
    Bourbaki, N.: Eléments de Mathématique, Algèbre commutative, chap. 5/6 (fasc.XXX), Paris (1964)Google Scholar
  3. [3]
    Delon, F.: Quelques propriétés des corps valués en théorie des modèles. Thèse Paris VII (1981)Google Scholar
  4. [4]
    Endler, O.: Valuation theory, Berlin-Heidelberg-New York (1972)Google Scholar
  5. [5]
    Ershov, Ju.L.: Decision problems and constructivizable models. Nauka, Moscow (1980) (Russian)Google Scholar
  6. [6]
    Huppert, B.: Endliche Gruppen I, Berlin-Heidelberg-New York (1967)Google Scholar
  7. [7]
    Kaplansky, I.: Maximal fields with valuations, Duke Math. Journ. 9, 303–321 (1942)Google Scholar
  8. [8]
    Krull, W.: Allgemeine Bewertungstheorie, J. reine angew. Math. 167, 160–196 (1931)Google Scholar
  9. [9]
    Neukirch, J.: Zur Verzweigungstheorie der allgemeinen Krullschen Bewertungen, Abh. Math. Sem. Univ. Hamburg 32, 207–215 (1968)Google Scholar
  10. [10]
    Ostrowski, A.: Untersuchungen zur arithmetischen Theorie der Körper, Math. Zeitschr. 39, 269–404 (1935)Google Scholar
  11. [11]
    Pank, M.: Beiträge zur reinen und angewandten Bewertungstheorie, Dissertation Heidelberg (1976)Google Scholar
  12. [12]
    Potthoff, K.: Einführung in die Modelltheorie und ihre Anwendungen, Darmstadt (1981)Google Scholar
  13. [13]
    Prestel, A.-Roquette, P.: Formally p-adic fields, Springer Lecture Notes in Math. 1050 (1984)Google Scholar
  14. [14]
    Robinson, A.: Nonstandard Arithmetic, Bull. Amer. Math. Soc. 73, 818–843 (1967)=Selected papers vol.2, 132–157 (1979)Google Scholar
  15. [15]
    Roquette, P.: Some Tendencies in Contemporary Algebra, in: W. Jäger et al. (ed.): Perspectives in Mathematics. Anniversary of Oberwolfach 1984, 393–422, Basel (1984)Google Scholar
  16. [16]
    Serre, J.P.: Cohomologie Galoisienne, 3rd ed. Springer Lecture Notes in Math. 5 (1965)Google Scholar
  17. [17]
    Schilling, O.F.G.: The theory of valuations, Math. Surveys A.M.S. (1950)Google Scholar
  18. [18]
    Steinitz, E.: Algebraische Theorie der Körper, Journ. reine angew. Math. Nr. 137, 167–309 (1910)=reprint Chelsea (1950)Google Scholar
  19. [19]
    Whaples, G.: Galois cohomology of additive polynomials and n-th power mappings of fields, Duke Math. Journ. 24, 143–150 (1957)Google Scholar
  20. [20]
    Zassenhaus, H.: The theory of groups, 2nd ed. New York (1958)Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Franz -Viktor Kuhlmann
    • 1
  • Matthias Pank
    • 1
  • Peter Roquette
    • 1
  1. 1.Mathematisches InstitutUniversität HeidelbergHeidelberg

Personalised recommendations