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Annali di Matematica Pura ed Applicata

, Volume 192, Issue 6, pp 1099–1114 | Cite as

Levels and sublevels of algebras obtained by the Cayley–Dickson process

Article

Abstract

In this paper, we generalize the concepts of level and sublevel of a composition algebra to algebras obtained by the Cayley–Dickson process and we will show that, in the case of level for algebras obtained by the Cayley–Dickson process, the situation is the same as for the integral domains, proving that for any positive integer n, there is an algebra A obtained by the Cayley–Dickson process with the norm form anisotropic over a suitable field, which has the level \({n \in \mathbb{N}-\{0\}}\) .

Keywords

Cayley–Dickson process Division algebra Level and sublevel of an algebra 

Mathematics Subject Classification (2000)

17A35 17A20 17A75 17A45 

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References

  1. 1.
    Brown R.B.: On generalized Cayley–Dickson algebras. Pac. J. Math. 20(3), 415–422 (1967)CrossRefMATHGoogle Scholar
  2. 2.
    Dai Z.D., Lam T.Y., Peng C.K.: Levels in algebra and topology. Bull. Am. Math. Soc. 3, 845–848 (1980)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Flaut C.: Isotropy of some quadratic forms and its applications on levels and sublevels of algebras. J. Math. Sci. Adv. Appl. 12(2), 97–117 (2011)MathSciNetMATHGoogle Scholar
  4. 4.
    Hoffman D.W.: Levels of quaternion algebras. Archiv der Mathematik 90(5), 401–411 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hoffman D.W.: Levels and sublevels of quaternion algebras. Math. Proc. R. Ir. Acad. 110A((1), 95–98 (2010)Google Scholar
  6. 6.
    Karpenko N.A., Merkurjev A.S.: Essential dimension of quadratics. Inventiones Mathematicae 153, 361–372 (2003)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Knebusch M.: Generic splitting of quadratic forms I. Proc. Lond. Math. Soc. 33, 65–93 (1976)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Laghribi A., Mammone P.: On the level of a quaternion algebra. Commun. Algebra 29(4), 1821–1828 (2001)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Lewis D.W.: Levels and sublevels of division algebras. Proc. R. Ir. Acad. Sect. A 87(1), 103–106 (1987)MATHGoogle Scholar
  10. 10.
    O’Shea J.: Bounds on the levels of composition algebras. Math. Proc. R. Ir. Acad. 110A(1), 21–30 (2010)MathSciNetMATHGoogle Scholar
  11. 11.
    O’Shea J.: Sums of squares in certain quaternion and octonion algebras. C. R. Acad. Sci. Paris Sèr. I Math. 349, 239–242 (2011)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Pfister A.: Zur Darstellung von-I als Summe von quadraten in einem Körper. J. Lond. Math. Soc. 40, 159–165 (1965)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Pumplün S.: Sums of squares in octonion algebras. Proc. Am. Math. Soc. 133, 3143–3152 (2005)CrossRefMATHGoogle Scholar
  14. 14.
    Schafer R.D.: An Introduction to Nonassociative Algebras. Academic Press, New York, NY (1966)MATHGoogle Scholar
  15. 15.
    Schafer R.D.: On the algebras formed by the Cayley–Dickson process. Am. J. Math. 76, 435–446 (1954)CrossRefMATHGoogle Scholar
  16. 16.
    Scharlau W.: Quadratic and Hermitian Forms. Springer-Verlag, Berlin (1985)CrossRefMATHGoogle Scholar
  17. 17.
    Tignol J.-P., Vast N.: Representation de −1 comme somme de carrè dans certain algébres de quaternions. C. R. Acad. Sci. Paris Sèr. Math. 305 13, 583–586 (1987)MathSciNetGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag 2012

Authors and Affiliations

  1. 1.“Ovidius” University of ConstanţaConstanţaRomânia

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