Skip to main content

Definable Valuations Induced by Definable Subgroups

  • Chapter
  • First Online:
Groups, Modules, and Model Theory - Surveys and Recent Developments

Abstract

In his paper Definable Valuations (1994) Koenigsmann shows that every field that admits a t-henselian topology is either real closed or separably closed or admits a definable valuation inducing the t-henselian topology. To show this Koenigsmann investigates valuation rings induced by certain (definable) subgroups of the field. The aim of this paper, based on the author’s PhD thesis (Dupont, PhD thesis, University of Konstanz, 2015), is to look at the methods used in Koenigsmann (Definable Valuations, 1994) in greater detail and Koenigsmann (Definable Henselian Valuations, J. Symb. Log. 80(01):85–99, 2015).

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. W. Anscombe, J. Koenigsmann, An existential 0-definition of F q [[t]] in F q ((t)). J. Symb. Log. 79(04), 1336–1343 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. J. Ax, On the undecidability of power series fields. Proc. Am. Math. Soc. 16, 846 (1965)

    MathSciNet  MATH  Google Scholar 

  3. Z. Chatzidakis, M. Perera, A criterion for p-henselianity in characteristic p (2015), http://arxiv.org/pdf/1509.04535v1.pdf

  4. R. Cluckers, J. Derakhshan, E. Leenknegt, A. Macintyre, Uniformly defining valuation rings in Henselian valued fields with finite or pseudo-finite residue fields. Ann. Pure Appl. Logic 164(12), 1236–1246 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. K. Dupont, V-Topologien auf Körpererweiterungen, KOPS (2010), http://kops.ub.uni-konstanz.de/handle/urn:nbn:de:bsz:352-208137

  6. K. Dupont, Definable valuations on NIP fields, PhD thesis, University of Konstanz (2015)

    Google Scholar 

  7. K. Dupont, A. Hasson, S. Kuhlmann, Definable valuations on NIP fields (in preparation)

    Google Scholar 

  8. H. Dürbaum, H.-J. Kowalski, Arithmetische Kennzeichnung von Körpertopologien. J. Reine Angew. Math. 191, 135–152 (1953)

    MathSciNet  MATH  Google Scholar 

  9. A.J. Engler, A. Prestel, Valued Fields (Springer, Berlin, 2005)

    MATH  Google Scholar 

  10. A. Fehm, Existential 0-definability of henselian valuation rings. J. Symb. Log. 80(1) (2015), arXiv 1307.1956

    Google Scholar 

  11. A. Fehm, A. Prestel, Uniform definability of Henselian valuation rings in the Macintyre language. Bull. Lond. Math. Soc. 47, 693–703 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. F. Jahnke, J. Koenigsmann, Definable Henselian valuations. J. Symb. Log. 80(01), 85–99 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. F. Jahnke, J. Koenigsmann, Uniformly defining p-henselian valuations. Ann. Pure Appl. Logic 166 741–754 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. F. Jahnke, P. Simon, E. Walsberg, dp-Minimal valued fields, available on ArXiv:http://arxiv.org/abs/1507.03911 (2015)

  15. W. Johnson, On dp-minimal fields, available on ArXiv:http://arxiv.org/abs/1507.02745 (2015)

  16. J. Koenigsmann, Definable valuations, Seminaire Structures algébriques ordonnées Paris VII, ed. by F. Delon, M. Dickmann, D. Gondard (1994)

    Google Scholar 

  17. J. Koenigsmann, p-Henselian fields. Manuscripta Math. 87, 89–99 (1995)

    Google Scholar 

  18. B. Poizat, A Course in Model Theory (Springer, Berlin, 2000)

    Book  MATH  Google Scholar 

  19. A. Prestel, C.N. Delzell, Positive Polynomials (Springer, Berlin, 2001)

    Book  MATH  Google Scholar 

  20. A. Prestel, M. Ziegler, Model theoretic methods in the theory of topological fields. J. Reine Angew. Math. 299/300, 318–341 (1978)

    Google Scholar 

  21. K. Tent, M. Ziegler, A Course in Model Theory (Cambridge University Press, Cambridge, 2012)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

I would like to thank Franziska Janke for pointing out the mistake in [16] as well as for several helpful discussions and comments on an early version of this work. Further I would like to thank Salma Kuhlmann and Assaf Hasson for great support and helpful advice while I was conducting the research as well as while I was writing the paper. Also, I would like to thank the referee for thoroughly reading the paper and making some helpful comments.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Katharina Dupont .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Dupont, K. (2017). Definable Valuations Induced by Definable Subgroups. In: Droste, M., Fuchs, L., Goldsmith, B., Strüngmann, L. (eds) Groups, Modules, and Model Theory - Surveys and Recent Developments . Springer, Cham. https://doi.org/10.1007/978-3-319-51718-6_5

Download citation

Publish with us

Policies and ethics