Skip to main content
SpringerLink
Log in

Subforms of norm forms of octonion fields

  • Published:
Archiv der Mathematik Aims and scope Submit manuscript

Abstract

We characterize the forms that occur as restrictions of norm forms of octonion fields. The results are applied to forms of types E\(_6\), E\(_7\), and E\(_8\) and to positive definite forms over fields that allow a unique non-split octonion algebra, e.g., the field of rational numbers.

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. A. Blunck, N. Knarr, B. Stroppel, and M. J. Stroppel, Groups of similitudes generated by octonions, Preprint 2017-007, Fachbereich Mathematik, Universität Stuttgart, Stuttgart, 2017, http://www.mathematik.uni-stuttgart.de/preprints/downloads/2017/2017-007.pdf.

  2. T. De Medts, A characterization of quadratic forms of type \(E_6,\ E_7\), and \(E_8\), J. Algebra 252 (2002), 394–410.

    Article  MathSciNet  MATH  Google Scholar 

  3. I. Kaplansky, Quadratic forms, J. Math. Soc. Japan 5 (1953), 200–207.

    Article  MathSciNet  MATH  Google Scholar 

  4. T. Y. Lam, Introduction to Quadratic Forms over Fields, Graduate Studies in Mathematics, 67, American Mathematical Society, Providence, RI, 2005.

  5. S. Lang, Algebra, Addison-Wesley Publishing Company Advanced Book Program, Reading, MA, 1984.

  6. H. Salzmann, T. Grundhöfer, H. Hähl, and R. Löwen, The Classical Fields, Encyclopedia of Mathematics and its Applications 112, Cambridge University Press, Cambridge, 2007.

    MATH  Google Scholar 

  7. C. Siegel, Darstellung total positiver Zahlen durch Quadrate, Math. Z. 11 (1921), 246–275.

    Article  MathSciNet  MATH  Google Scholar 

  8. T. A. Springer and F. D. Veldkamp, Octonions, Jordan Algebras and Exceptional Groups, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2000.

  9. J. Tits and R. M. Weiss, Moufang Polygons, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002.

  10. M. Zorn, Alternativkörper und quadratische Systeme, Abh. Math. Sem. Univ. Hamburg 9 (1933), 395–402.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors thank an anonymous referee for pointing out that 4.3.3 follows in the context of our present results.

Author information

Authors and Affiliations

  1. LExMath, Fakultät für Mathematik und Physik, Universität Stuttgart, 70550, Stuttgart, Germany

    Norbert Knarr & Markus J. Stroppel

Authors
  1. Norbert Knarr
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Markus J. Stroppel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Markus J. Stroppel.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Knarr, N., Stroppel, M.J. Subforms of norm forms of octonion fields. Arch. Math. 110, 213–224 (2018). https://doi.org/10.1007/s00013-017-1129-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00013-017-1129-x

Mathematics Subject Classification

Keywords

Search

Navigation