Skip to main content
SpringerLink
Log in

On PAC extensions and scaled trace forms

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

Any non-degenerate quadratic form over a Hilbertian field (e.g., a number field) is isomorphic to a scaled trace form. In this work we extend this result to more general fields, in particular, prosolvable and prime-to-p extensions of a Hilbertian field. The proofs are based on the theory of PAC extensions.

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

References

  1. L. Bary-Soroker, Dirichlet’s theorem for polynomial rings, Proceedings of the American Mathematical Society 137 (2009), 73–83.

    Article  MATH  MathSciNet  Google Scholar 

  2. L. Bary-Soroker and M. Jarden, PAC fields over finitely generated fields, Mathematische Zeitschrift 260(2) (2008),329–334.

    Article  MathSciNet  Google Scholar 

  3. P. E. Conner and R. Perlis, A Survey of Trace Forms of Algebraic Number Fields, World Scientific Publ. Singapore, 1984.

    MATH  Google Scholar 

  4. M. Epkenhans, On trace forms of algebraic number fields, Archiv der Mathematik 60 (1993), 527–529.

    Article  MATH  MathSciNet  Google Scholar 

  5. M. D. Fried and M. Jarden, Field Arithmetic, second edition, revised and enlarged by Moshe Jarden, Ergebnisse der Mathematik (3) 11, Springer-Verlag, Heidelberg, 2005.

    MATH  Google Scholar 

  6. C. Hermite, Remarques sur le theoreme de Sturm, Comptes Rendus Mathematique. Academie des Sciences. Paris 36 (1853), 294–297.

    Google Scholar 

  7. M. Jarden and A. Razon, Pseudo algebraically closed fields over rings, Israel Journal of Mathematics 86 (1994), 25–59.

    Article  MATH  MathSciNet  Google Scholar 

  8. M. Krüskemper, Algebraic number field extensions with prescribed trace form, Journal of Number Theory 40 (1992), 120–124.

    Article  MATH  MathSciNet  Google Scholar 

  9. M. Krüskemper and W. Scharlau, On trace forms of Hilbertian fields, Commentarii Mathematici Helvetici 63 (1988), 296–304.

    Article  MATH  MathSciNet  Google Scholar 

  10. W. Scharlau, On trace forms of algebraic number fields, Mathematische Zeitschrift 196 (1987), 125–127.

    Article  MATH  MathSciNet  Google Scholar 

  11. J. P. Serre, L’invariant de Witt de la forme Tr(x 2), Commentarii Mathematici Helvetici 59 (1984), 651–676.

    Article  MATH  MathSciNet  Google Scholar 

  12. J. J. Sylvester, On a theory of the Syzygetic relations of two rational integral functions, comprising an application to the theory of Sturm’s functions, and that of the greatest algebraical common measure, Philosophical transactions of the Royal Society of London 143 (1853), 407–548.

    Article  Google Scholar 

  13. O. Taussky-Todd, The discriminant of an algebraic number field, Journal of the London Mathematical Society 43 (1968), 152–154.

    Article  Google Scholar 

  14. W. C. Waterhouse, Scaled trace forms over number fields, Archiv der Mathematik Birkäuser, Basel 47 (1986), 229–231.

    Google Scholar 

  15. E. Witt, Theorie der quadratischen Formen in beliebigen Körpern, Journal für die Reine und Angewandte Mathematik 176 (1937), 31–44.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. The Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv, 69978, Israel

    Lior Bary-Soroker & Dubi Kelmer

Authors
  1. Lior Bary-Soroker
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dubi Kelmer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lior Bary-Soroker.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bary-Soroker, L., Kelmer, D. On PAC extensions and scaled trace forms. Isr. J. Math. 175, 113–124 (2010). https://doi.org/10.1007/s11856-010-0004-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-010-0004-2

Keywords

Search

Navigation