Skip to main content
SpringerLink
Log in

Fields with two incomparable henselian valuation rings

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

Let K be a field and C, C' be two incomparable valuation rings of the separable closure of K, Theorem 1.2 states that the intersection of the decomposition groups of C, C', with respect to K, is precisely the inertia group of the composition ring C·C'. We apply this theorem in the study of two special cases of valued fields (L,B). In the first case, B is henselian and there is a subfield K of L such that L|K is a normal extension and B ∩ K is not henselian. The second case is that in which B has exactly two prolongations in the separable closure of L. We call these rings semihenselian rings, and they are characterized through Theorems 2.6 and 2.12.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Bibliography

  1. Bourbaki, N.: Algèbre Commutative. Paris: Hermann (1964)

    Google Scholar 

  2. Endler, O.: A Note On Henselian Valuation Rings. Canadian math. Bull. 11, 185–189 (1968).

    Google Scholar 

  3. Endler, O.: Valuation Theory. Berlin-Heidelberg-New York: Springer-Verlag (1972).

    Google Scholar 

  4. Endler, O.: On Henselizations of Valued Fields. Bol. Soc. Bras. Mat.4, 97–109 (1973)

    Google Scholar 

  5. Endler, O.: Finite Separable Field Extension With Prescribed Extensions of Valuations. Manuscripta math.17, 383–408 (1975).

    Google Scholar 

  6. Endler, O., Engler, A.J.: Fields With Henselian Valuation Rings. Math. Z.152, 191–193 (1977).

    Google Scholar 

  7. Lang, S.: The Theory of Real Places. Ann. of Math., II. Ser.57, 378–391 (1953).

    Google Scholar 

  8. Prestel, A.: Lectures on Formally Real Fields. Rio de Janeiro: IMPA (1975).

    Google Scholar 

  9. Ribenboim, P.: Corps Maximaux et Complets par des Valuation de Krull. Math. Z.69, 466–479 (1958).

    Google Scholar 

  10. Ribenboim, P.: Theorie des Valuations. 20 édition. Montreal: Les Presses de l' Université de Montreal (1968).

    Google Scholar 

  11. Ribenboim, P.: A Short Note On Henselian Fields. Math. Ann.173, 83–88 (1967).

    Google Scholar 

  12. Rim, D.: Relatively Conplete Fields. Duke math. J.24, 197–200 (1957).

    Google Scholar 

  13. Wright, M.: On The Number of Prolongations of a Finite Rank Valuation. Canadian J. Math.23 553–556 (1971).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. IMECC-UNICAMP, 13.100 Campinas, Caixa Postal 1170, Brazil

    Antonio J. Engler

Authors
  1. Antonio J. Engler
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This paper is part of author's doctoral dissertation. Financial support for this research was provided by CNPq (National Research Council) and by Universidade Estadual de Campinas.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Engler, A.J. Fields with two incomparable henselian valuation rings. Manuscripta Math 23, 373–385 (1978). https://doi.org/10.1007/BF01167696

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01167696

Keywords

Search

Navigation