Skip to main content
SpringerLink
Log in

The weak isotropy of quadratic forms over field extensions

  • Published:
Manuscripta Mathematica Aims and scope Submit manuscript

Abstract

The weak isotropy index (or equivalently, sublevel) of arbitrary quadratic forms is studied. Its relationship to the level of a form is investigated. The problem of determining the set of values of the weak isotropy index of a form as it ranges over field extensions is addressed, with both admissible and inadmissible numbers being determined. An analogous investigation with respect to the level of a form is also undertaken. A treatment of forms for which the above invariants coincide concludes this article, with some recently-raised questions being resolved.

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. Becher K.J.: Minimal weakly isotropic forms. Math. Z. 252(1), 91–102 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. Berhuy G., Grenier-Boley N., Mahmoudi M.G.: Sums of values represented by a quadratic form. Manuscr. Math. 140(3-4), 531–556 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  3. Elman R., Lam T.Y., Wadsworth A.R.: Orderings under field extensions. J. Reine Angew. Math. 306, 7–27 (1979)

    MATH  MathSciNet  Google Scholar 

  4. Gentile E.R., Shapiro D.B.: Conservative quadratic forms. Math. Z. 163, 15–23 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  5. Hoffmann D.W.: Isotropy of quadratic forms over the function field of a quadric. Math. Z. 220(3), 461–476 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  6. Karpenko N.A.: On the first Witt index of quadratic forms. Invent. Math. 153, 455–462 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Karpenko N.A., Merkurjev A.S.: Essential dimension of quadrics. Invent. Math. 153, 361–372 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. M. Knebusch: Generic splitting of quadratic forms, I. Proc. Lond. Math. Soc. (3) 33, 65–93 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  9. Lam T. Y.: Introduction to Quadratic Forms over Fields. American Mathematical Society, Providence (2005)

    MATH  Google Scholar 

  10. O’Shea, J.: Products of Pfister forms (2014, in preparation)

  11. Pfister, A.: Quadratic forms with applications to algebraic geometry and topology. In: London Mathematical Society Lecture Note, vol. 217. Cambridge University Press, Cambridge (1995)

  12. Wadsworth A.R., Shapiro D.B.: On multiples of round and Pfister forms. Math. Z. 157, 53–62 (1977)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, National University of Ireland Maynooth, Maynooth, Co. Kildare, Ireland

    James O’Shea

Authors
  1. James O’Shea
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to James O’Shea.

Rights and permissions

Reprints and permissions

About this article

Cite this article

O’Shea, J. The weak isotropy of quadratic forms over field extensions. manuscripta math. 145, 143–161 (2014). https://doi.org/10.1007/s00229-014-0671-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-014-0671-0

Mathematics Subject Classification

Search

Navigation