Skip to main content
SpringerLink
Log in

A normal form and schemes of quadratic forms

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

We present a solution of the problem of construction of a normal diagonal form for quadratic forms over a local principal ideal ring R = 2R with a QF-scheme of order 2. We give a combinatorial representation for the number of classes of projective congruence quadrics of the projective space over R with nilpotent maximal ideal. For the projective planes, the enumeration of quadrics up to projective equivalence is given; we also consider the projective planes over rings with nonprincipal maximal ideal. We consider the normal form of quadratic forms over the field of p-adic numbers. The corresponding QF-schemes have order 4 or 8. Some open problems for QF-schemes are mentioned. The distinguished finite QF-schemes of local and elementary types (of arbitrarily large order) are realized as the QF-schemes of a field.

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. E. Artin, Geometric Algebra, Interscience, New York (1959).

    Google Scholar 

  2. R. Baeza, “On the classification of quadratic forms over semilocal rings,” Bull. Soc. Math. France, Suppl., Mém., 59, Colloque sur les Formes Quadratiques II (Montpellier, 1977), 7–10 (1979).

    MathSciNet  Google Scholar 

  3. R. Baeza, Quadratic Forms over Semilocal Rings, Lect. Notes Math., Vol. 655, Springer, Berlin (1978).

    MATH  Google Scholar 

  4. R. Baeza, “The norm theorem for quadratic forms over a field of characteristic 2,” Comm. Algebra, 18, No. 5, 1337–1348 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  5. K. J. Becher, “On the number of square classes of a field of finite level,” Documenta Math., Extra Volume, Quadratic Forms LSU, 65–84 (2001).

  6. K. J. Becher and D. W. Hoffmann, “Symbol lengths in Milnor K-theory,” Homology Homotopy Appl., 6, No. 1, 17–31 electronic only (2004).

    MATH  MathSciNet  Google Scholar 

  7. W. Benz, Vorlesungen über Geometrie der Algebren, Springer, Berlin (1973).

    MATH  Google Scholar 

  8. N. Bourbaki, Éléments de mathématique, XXIII. Première partie: Les structures fondamentales de l’analyse. Livre II: Algèbre. Chapitre 8: Modules et anneaux semi-simples, Actualités Sci. Ind., Vol. 1261, Hermann, Paris (1958).

    Google Scholar 

  9. J. W. S. Cassels, Rational Quadratic Forms, London Math. Soc. Monographs, Vol. 13, Academic Press, London (1978).

    MATH  Google Scholar 

  10. J.-L. Colliot-Thélène, “Formes quadratiques sur les anneaux semi-locaux réguliers,” Bull. Soc. Math. France, Suppl., Mém., 59, Colloque sur les Formes Quadratiques II (Montpellier, 1977), 13–31 (1979).

    Google Scholar 

  11. A. Corduneanu, “An elementary proof of the inertial law for real quadratic forms,” Gaz. Mat., Ser. Inf. Stiint. Perfect. Metod, 15, No. 1, 41–46 (1997).

    MATH  Google Scholar 

  12. D. R. Estes and R. M. Guralnick, “A stable range for quadratic forms over commutative rings,” J. Pure Appl. Algebra, 120, No. 3, 255–280 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  13. E. A. Hornix, “Totally indefinite quadratic forms over formally real fields,” Indag. Math., 47, No. 3, 305–312 (1985).

    MathSciNet  Google Scholar 

  14. M. Kula, “Fields and quadratic form schemes,” Ann. Math. Sil., 1(13), 7–22 (1985).

    MathSciNet  Google Scholar 

  15. M. Kula, “Counting Witt rings,” J. Algebra, 206, No. 2, 568–587 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  16. T. Y. Lam, Algebraic Theory of Quadratic Forms, Benjamin/Cummings Publ., Reading (1980).

    MATH  Google Scholar 

  17. V. M. Levchuk and O. A. Starikova, “Quadrics and symmetric forms of modules over a local ring of principal ideals,” in: Int. Algebraic Conf., Abstracts [in Russian], Moscow State University, Moscow (2004), pp. 86–88.

    Google Scholar 

  18. V. M. Levchuk and O. A. Starikova, “Quadratic forms of projective spaces over rings,” Sb. Math., 197, No. 6, 887–899 (2006).

    Article  Google Scholar 

  19. M. Marshall, “The elementary type conjecture in quadratic form theory,” in: R. Baeza et al., eds., Algebraic and Arithmetic Theory of Quadratic Forms. Proc. of the Int. Conf., Universidad de Talca, Talca and Pucon, Chile, December 11–18, 2002, Contemp. Math., Vol. 344, Amer. Math. Soc., Providence (2004), pp. 275–293.

    Google Scholar 

  20. Yu. I. Merzlyakov, ed., Automorphisms of the Classical Groups [in Russian], Mir, Moscow (1976).

    Google Scholar 

  21. J. Milnor and D. Husemoller, Symmetric Bilinear Forms, Ergebnisse Math. ihrer Grenzgebiete, Vol. 73, Springer, New York (1973).

    MATH  Google Scholar 

  22. M. Ojanguren and R. Sridharan, “A note on the fundamental theorem of projective geometry,” Comment Math. Helv., 44, 310–315 (1969).

    Article  MATH  MathSciNet  Google Scholar 

  23. W. Scharlau, Quadratic and Hermitian Forms, Springer, Berlin (1985).

    MATH  Google Scholar 

  24. O. A. Starikova, “Listing quadrics of projective planes and spaces over local rings of principal ideals,” in: Algebra and Model Theory [in Russian], Novosibirsk State Technical University, Novosibirsk (2003), pp. 110–115.

    Google Scholar 

  25. O. A. Starikova, Listing Quadrics and Symmetric Forms of Modules over Local Rings [in Russian], Ph.D. Thesis, Northern International University, Magadan (2004).

    Google Scholar 

  26. V. V. Vishnevskii, B. A. Rozenfel’d, and A. P. Shirokov, “On the development of the geometry of spaces over algebras,” Sov. Math. (Iz. VUZ), 28, No. 7, 46–54 (1984).

    MathSciNet  Google Scholar 

  27. V. V. Vishnevskii, A. P. Shirokov, and V. V. Shurygin, Spaces over Algebras [in Russian], Kazan. Gos. Univ., Kazan (1985).

    Google Scholar 

  28. B. L. van der Waerden, Algebra, Vols. I, II, Springer, New York (1991).

    Google Scholar 

  29. J. L. Yucas, “A classification theorem for quadratic forms over semilocal rings,” Ann. Math. Sil., 2, No. 14, 7–12 (1986).

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Krasnoyarsk State University, Krasnoyarsk, Russia

    V. M. Levchuk

  2. Northern International University, Magadan, Russia

    O. A. Starikova

Authors
  1. V. M. Levchuk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. O. A. Starikova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. M. Levchuk.

Additional information

__________

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 1, pp. 161–178, 2007.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Levchuk, V.M., Starikova, O.A. A normal form and schemes of quadratic forms. J Math Sci 152, 558–570 (2008). https://doi.org/10.1007/s10958-008-9078-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-008-9078-3

Keywords

Search

Navigation