Quadratic and Higher Degree Forms pp 255-298

Part of the Developments in Mathematics book series (DEVM, volume 31) | Cite as

Identifying the Matrix Ring: Algorithms for Quaternion Algebras and Quadratic Forms

Chapter

Abstract

We discuss the relationship between quaternion algebras and quadratic forms with a focus on computational aspects. Our basic motivating problem is to determine if a given algebra of rank 4 over a commutative ring R embeds in the 2 ×2-matrix ring M2(R) and, if so, to compute such an embedding. We discuss many variants of this problem, including algorithmic recognition of quaternion algebras among algebras of rank 4, computation of the Hilbert symbol, and computation of maximal orders.

Key words and Phrases

Quadratic forms Quaternion algebras Maximal orders Algorithms Matrix ring Number theory 

Mathematics Subject Classification (2010):

Primary 11R52 Secondary 11E12 

References

  1. 1.
    Manindra Agrawal, Neeraj Kayal, and Nitin Saxena, PRIMES is in P, Ann. of Math. (2) 160 (2004), no. 2, 781–793.Google Scholar
  2. 2.
    Leonard M. Adleman, Carl Pomerance, and Robert S. Rumely, On distinguishing prime numbers from composite numbers, Ann. of Math. (2) 117 (1983), no. 1, 173–206.Google Scholar
  3. 3.
    Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), vol. 3–4, 235–265.Google Scholar
  4. 4.
    J. A. Buchmann and H. W. Lenstra, Jr., Approximating rings of integers in number fields, J. Théor. Nombres Bordeaux 6 (1994), no. 2, 221–260.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    A. L. Chistov, The complexity of the construction of the ring of integers of a global field, Soviet Math. Dokl. 39 (1989), no. 3, 597–600.MathSciNetMATHGoogle Scholar
  6. 6.
    Henri Cohen, Computational algebraic number theory, Grad. Texts in Math., vol. 193, Springer, Berlin, 2000.Google Scholar
  7. 7.
    Henri Cohen, Advanced topics in computational algebraic number theory, Grad. Texts in Math., vol. 193, Springer, Berlin, 2000.Google Scholar
  8. 8.
    J. E. Cremona and D. Rusin, Efficient solution of rational conics, Math. Comp. 72 (2003), no. 243, 1417–1441.MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Harold Davenport, Multiplicative number theory, 3rd. ed., Graduate texts in mathematics, vol. 74, Springer-Verlag, Berlin, 2000.Google Scholar
  10. 10.
    Leonard Eugene Dickson, Algebras and their arithmetics, Dover, New York, 1960.Google Scholar
  11. 11.
    Carsten Friedrichs, Berechnung von Maximalordnungen uber Dedekindringen, Ph. D. dissertation, Technischen Universität Berlin, 2000.Google Scholar
  12. 12.
    A. Fröhlich, Local fields, in Algebraic number theory, J.W.S. Cassels and A. Fröhlich, eds., Thompson Book Company, Washington, 1967, 1–41.Google Scholar
  13. 13.
    Joachim von zur Gathen and Jürgen Gerhard, Modern computer algebra, 2nd edition, Cambridge University Press, Cambridge, 2003.Google Scholar
  14. 14.
    Florian Hess, Computing Riemann-Roch spaces in algebraic function fields and related topics, J. Symbolic Comput. 33 (2002), no. 4, 425–445.MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Gábor Ivanyos and Ágnes Szántó, Lattice basis reduction for indefinite forms and an application, Proceedings of the 5th Conference on Formal Power Series and Algebraic Combinatorics (Florence, 1993), Discrete Math. 153 (1996), no. 1–3, 177–188.MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Gábor Ivanyos and Lajos Rónyai, Finding maximal orders in semisimple algebras over \(\mathbb{Q}\), Comput. Complexity 3 (1993), no. 3, 245–261.MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Nathan Jacobson, Finite-dimensional division algebras over fields, Springer-Verlag, Berlin, 1996.MATHCrossRefGoogle Scholar
  18. 18.
    Markus Kirschmer and John Voight, Algorithmic enumeration of ideal classes for quaternion orders, SIAM J. Comput. (SICOMP) 39 (2010), no. 5, 1714–1747.MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Max-Albert Knus, Quadratic forms, Clifford algebras and spinors, Seminários de Matemática, 1, Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Ciência da Computaç ã o, Campinas, 1988.Google Scholar
  20. 20.
    Max-Albert Knus, Alexander Merkurjev, and Jean-Pierre Tignol, The book of involutions, American Math. Soc. Colloquium Publications, vol. 44, AMS, Providence, RI, 1998.Google Scholar
  21. 21.
    T.Y. Lam, A first course in noncommutative rings, 2nd ed., Graduate texts in mathematics, vol. 131, American Math. Soc., Providence, 2001.Google Scholar
  22. 22.
    H.W. Lenstra, Jr., Algorithms in algebraic number theory, Bull. Amer. Math. Soc. (N.S.) 26 (1992), no. 2, 211–244.Google Scholar
  23. 23.
    H. W. Lenstra, Jr., Computing Jacobi symbols in algebraic number fields, Nieuw Arch. Wisk. (4) 13 (1995), no. 3, 421–426.Google Scholar
  24. 24.
    Gabriele Nebe and Allan Steel, Recognition of division algebras, J. Algebra 322 (2009), no. 3, 903–909.MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Jürgen Neukirch, Algebraic number theory, Grundlehren der mathematischen Wissenschaften, vol. 322, Springer-Verlag, Berlin, 1999.Google Scholar
  26. 26.
    O.Timothy O’Meara, Introduction to quadratic forms, Classics in Mathematics, Springer-Verlag, Berlin, 2000.Google Scholar
  27. 27.
    Michael Pohst and Hans Zassenhaus, Algorithmic algebraic number theory, Revised reprint, Encyclopedia of Mathematics and its Applications, vol. 30, Cambridge University Press, Cambridge, 1997.Google Scholar
  28. 28.
    Irving Reiner, Maximal orders, Clarendon Press, Oxford, 2003.MATHGoogle Scholar
  29. 29.
    Lajos Rónyai, Zero divisors in quaternion algebras, J. Algorithms 9 (1988), 494–506.MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Lajos Rónyai, Algorithmic properties of maximal orders in simple algebras over \(\mathbb{Q}\), Comput. Complexity 2 (1992), no. 3, 225–243.MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Lajos Rónyai, Simple algebras are difficult, Proceedings, 19th ACM Symp. on Theory of Computing, 1990, 398–408.Google Scholar
  32. 32.
    Lajos Rónyai, Computing the structure of finite algebras, J. Symbolic Computation 9 (1990), 355–373.MATHCrossRefGoogle Scholar
  33. 33.
    Winfried Scharlau, Quadratic and Hermitian forms, Springer-Verlag, Berlin, 1985.MATHCrossRefGoogle Scholar
  34. 34.
    Viggo Stoltenberg-Hansen and John V. Tucker, Computable rings and fields, Handbook of computability theory, ed. Edward R. Griffor, North-Holland, Amsterdam, 1999, 336–447.Google Scholar
  35. 35.
    Dénis Simon, Equations dans les corps de nombres et discriminants minimaux, thèse, Universit Bordeaux I, 1998.Google Scholar
  36. 36.
    Dénis Simon, Solving quadratic equations using reduced unimodular quadratic forms, Math. Comp. 74 (2005), no. 251, 1531–1543.MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Christiaan van de Woestijne, Deterministic equation solving over finite fields, ISSAC’05, ACM, New York, 2005, 348–353.Google Scholar
  38. 38.
    Marie-France Vignéras, Arithmétique des algèbres de quaternions, Lecture notes in mathematics, vol. 800, Springer, Berlin, 1980.Google Scholar
  39. 39.
    John Voight, Quadratic forms and quaternion algebras: Algorithms and arithmetic, Ph.D. thesis, University of California, Berkeley, 2005.Google Scholar
  40. 40.
    John Voight, Rings of low rank with a standard involution, Illinois J. Math. 55 (2011), no. 3, 1135–1154.MathSciNetMATHGoogle Scholar
  41. 41.
    John Voight, Characterizing quaternion rings over an arbitrary base, J. Reine Angew. Math. 657 (2011), 113–134MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of VermontBurlingtonUSA

Personalised recommendations