Skip to main content

Sur les ordres de niveau 2n et sur une extension du 17ème probleme de Hilbert

Other Papers

  • 776 Accesses

Part of the Lecture Notes in Mathematics book series (LNM,volume 1524)

Abstract

Let K be a chainable field, which means a field admitting orderings of exact level 2n for any n≥1, from [G1] and [G2] we know that in such a field there exists α ∈ K such that α2 ∉ Σ K4; we then define the notion of α-chain of orderings and use it to obtain explicit formulas for orderings of exact level 2n in some special fields. We also obtain for some chain-closed fields a new extension of Hilbert's 17th problem.

Keywords

  • Holomorphy Ring
  • Chainable Field
  • Real Holomorphy Ring

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.

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

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   54.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   69.95
Price excludes VAT (Canada)
  • Compact, lightweight 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. E. Becker: "Hereditarily pythagorean fields and orderings of higher" types", I.M.P.A., Lectures Notes # 29 (1978), Rio de Janeiro.

    Google Scholar 

  2. E. Becker: "The real holomorphy ring and sums of 2n-th powers", in Géométrie Algébrique réelle, Lecture Notes in Math. # 959, p.138–181, Springer-Verlag 1982.

    Google Scholar 

  3. E. Becker, R. Berr, F. Delon et D. Gondard: "Hilbert's 17th problem and sums of 2n powers", preprint.

    Google Scholar 

  4. E. Becker et D. Gondard: "On rings admitting orderings and 2-primary chains of orderings of higher level", Manuscripta Mathematica, 69, 267–274, 1990.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. F. Delon et D. Gondard: "17ème problème de Hilbert au niveau n dans les corps chaîne-clos", The Journal of Symbolic Logic, vol. 56, # 3 Sept. 1991, p.853–861.

    CrossRef  MathSciNet  Google Scholar 

  6. M. Dickmann: "Couples d'ordres chaînables", Exposé au Séminaire "Structures algébriques ordonnées" (D.D.G.), Univ. paris VII 1987–88.

    Google Scholar 

  7. D. Gonard: "Théorie du premier ordre des corps chaînables et des corps chaîne-clos", C. R. Acad. Sc. Paris, Tome 304, # 16, 1987.

    Google Scholar 

  8. D. Gondard: "Chainable fields and real algabraic geometry", in Proceedings "Real Algebraic and Analytic Geometry" (Trento Oct. 88), Lectures Notes in Math. 1420, Springer-Verlag, 1990.

    Google Scholar 

  9. D. Gondard: "Kernels of chains and chainable fields", in Abstracts A.M.S., vol 10 #5 Oct. 1989.

    Google Scholar 

  10. D. Gondard: "Sur l'espace des places réelles d'un corps chaînable", preprint.

    Google Scholar 

  11. D. Gondard et P. Ribenboim: "Sur le 17ème problème de Hilbert", C. R. Acad. Sc. Paris, tome 277, 20-08-1973, p. 303–304.

    MathSciNet  MATH  Google Scholar 

  12. J. Harman: "Chains of higher level orderings", Contemporary Mathematics, vol. 8, 1982, pp. 141–174, A.M.S..

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. T. Y. Lam: "The theory of ordered fields", Proceedings of Alg. Conference, pp 1–152, M. Dekker (1980).

    Google Scholar 

  14. T. Y. Lam: "Orderings, Valuations and Quadratic Forms", C.B.M.S. regional conference, # 52, 1983, A.M.S..

    Google Scholar 

  15. K. McKenna: "New facts about Hilbert's 17th problem", Lecture notes in Math. 498, Springer-Verlag, pp. 220–230.

    Google Scholar 

  16. A. Prestel: "Lectures on formally real fields", I.M.P.A., Monografias de Matematica, # 22, 1975, Rio de Janeiro.

    Google Scholar 

  17. A. Prestel: "Sums of squares in fields", Bol. Soc. Brasil., Rio de Janeiro, 1975.

    Google Scholar 

  18. P. Ribenboim: "Arithmétique des corps", Herman, Paris 1972.

    MATH  Google Scholar 

  19. Zeng Guangxin: "A characterization of preordered fields with the weak Hilbert property", proc. of the A.M.S. 104 (1988), 335–342.

    CrossRef  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1992 Springer-Verlag

About this paper

Cite this paper

Gondard-Cozette, D. (1992). Sur les ordres de niveau 2n et sur une extension du 17ème probleme de Hilbert. In: Coste, M., Mahé, L., Roy, MF. (eds) Real Algebraic Geometry. Lecture Notes in Mathematics, vol 1524. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0084625

Download citation

  • DOI: https://doi.org/10.1007/BFb0084625

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-55992-4

  • Online ISBN: 978-3-540-47337-4

  • eBook Packages: Springer Book Archive