Polynomial time solutions of some problems of computational algebra
STOC '85 Proceedings of the seventeenth annual ACM symposium on Theory of computing
STOC '90 Proceedings of the twenty-second annual ACM symposium on Theory of computing
Decomposition of *-closed algebras in polynomial time
ISSAC '93 Proceedings of the 1993 international symposium on Symbolic and algebraic computation
Deciding finiteness of matrix groups in deterministic polynomial time
ISSAC '93 Proceedings of the 1993 international symposium on Symbolic and algebraic computation
Efficient decomposition of associative algebras
ISSAC '96 Proceedings of the 1996 international symposium on Symbolic and algebraic computation
Polynomial time algorithms for modules over finite dimensional algebras
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
Multiplicative equations over commuting matrices
Proceedings of the seventh annual ACM-SIAM symposium on Discrete algorithms
ISSAC '00 Proceedings of the 2000 international symposium on Symbolic and algebraic computation
Polynomial-time normalizers for permutation groups with restricted composition factors
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
Efficient decomposition of separable algebras
Journal of Symbolic Computation
Polynomial-time theory of matrix groups
Proceedings of the forty-first annual ACM symposium on Theory of computing
On the densest MIMO lattices from cyclic division algebras
IEEE Transactions on Information Theory
Construction methods for asymmetric and multiblock space-time codes
IEEE Transactions on Information Theory
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In this paper we address some algorithmic problems related to computations in finite-dimensional associative algebras over finite fields. Our starting point is the structure theory of finite-dimensional associative algebras. This theory determines, mostly in a nonconstructive way, the building blocks of these algebras. Our aim is to give polynomial time algorithms to find these building blocks, the radical and the simple direct summands of the radical-free part. The radical algorithm is based on a new, tractable characterisation of the radical. The algorithm for decomposition of semisimple algebras into simple ideals involves (and generalises) factoring polynomials over the ground field. Next, we study the problem of finding zero divisors in finite algebras. We show that thisproblem is in the same complexity class as the problem of factoring polynomials over finte fields. Applications include a polynomial time Las Vegas method to find a common invariant subspace of a set of linear transformations as well as an explicit isomorphism between a given finite simple algebra and a full matrix algebra over a finite field.