Abstract
We consider the boolean complexity of the decomposition of semi-simple algebras over finite fields and number fields.
We present new polynomial time algorithms for the decomposition of semi-simple algebras over these fields. Our algorithms are somewhat simpler than previous algorithms, and provide parallel reductions from semi-simple decomposition to the factorization of polynomials. As a consequence we obtain efficient parallel algorithms for the decomposition of semi-simple algebras over small finite fields. We also present efficient sequential and parallel algorithms for the decomposition of a simple algebra from a basis and a primitive idempotent. These will be applied in a subsequent paper to obtain Las Vegas polynomial time algorithms for the decomposition of matrix algebras over ℂ and ℝ.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. Babai andL. Rónyai, Computing irreducible representations of finite groups,Math. Comp. 55 (1990), 705–722.
S. J. Berkowitz, On computing the determinant in small parallel time using a small number of processors,Information Processing Letters 18 (1984), 147–150.
E. R. Berlekamp, Factoring polynomials over finite fields,Bell System Tech. J. 46 (1967), 1853–1859.
E. R. Berlekamp, Factoring polynomials over large finite fields,Math. Comp. 24 (1970), 713–735.
A. Borodin, J. von zur Gathen, andJ. Hopcroft, Fast parallel matrix and GCD computations,Information and Control 52 (1982), 241–256.
D. G. Cantor andH. Zassenhaus, A new algorithm for factoring polynomials over finite fields.Math. Comp. 36 (1981), 587–592.
C. W. Curtis andI. Reiner,Representation Theory of Finite Groups and Associative Algebras, Wiley, New York, 1962.
W. Eberly,Computations for algebras and group representations, Department of Computer Science Technical Report 225/89, University of Toronto, 1989.
W. Eberly, Decomposition of algebras over ℝ and ℂ,Computational Complexity 1 (1991), 207–230.
K. Friedl and L. Rónyai, Polynomial time solutions for some problems in computational algebra, inProc. 17th Ann. Symp. Theory of Computing, 1985, 153–162.
J. Gabriel, New methods for reduction of group representations using an extension of Schur's lemma,J. Math. Phys. 5 (1964), 494–504.
J. Gabriel, New methods for reduction of group representations II,J. Math. Phys. 9 (1968), 973–976.
J. Gabriel, New methods for reduction of group representations III,J. Math. Phys. 10 (1969), 1789–1795.
J. Gabriel, new methods for reduction of group representations IV,J. Math. Phys. 10 (1969), 1932–1934.
J. Gabriel, Numerical methods for reduction of group representations, inProc. 2nd ACM Symp. Symbolic and Algebraic Manipulation, 1971, 180–182.
J. von zur Gathen, Parallel algorithms for algebraic problems,SIAM J. Comput. 13 (1984), 802–824.
J. von zur Gathen, Irreducibility of multivariate polynomials,JCSS 31 (1985), 225–264.
S. Landau, Factoring polynomials over algebraic number fields,SIAM J. Comput. 14 (1985), 184–196.
A. K. Lenstra, H. W. Lenstra, andL. Lovász, Factoring polynomials with rational coefficients,Math. Ann. 261 (1982), 515–534.
K. Mulmuley, A fast parallel algorithm to compute the rank of a matrix over an arbitrary field,Combinatorica 7 (1987), 101–104.
R. S. Pierce,Associative Algebras, Springer-Verlag, New York, 1982.
M. Rabin, Probabilistic algorithms in finite fields,SIAM J. Comput. 9 (1980), 273–280.
L. Rónyai, Simple algebras are difficult, inProc. 19th Ann. Symp. Theory of Computing, 1987, 398–408.
L. Rónyai, Zero divisors in quaternion algebras,Journal of Algorithms 9 (1988), 494–506.
L. Rónyai, Algorithmic properties of maximal orders in simple algebras over ℚ, manuscript, 1991.
J. T. Schwartz, Fast probabilistic algorithms for verification of polynomial identities,JACM 27 (1980), 701–717.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Eberly, W. Decomposition of algebras over finite fields and number fields. Comput Complexity 1, 183–210 (1991). https://doi.org/10.1007/BF01272520
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01272520