Parallel computation for well-endowed rings and space-bounded probabilistic machines
Information and Control
Factoring polynominals over algebraic number fields
EUROCAL '83 Proceedings of the European Computer Algebra Conference on Computer Algebra
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
Computing the structure of finite algebras
Journal of Symbolic Computation - Special issue on computational algebraic complexity
Deciding finiteness of matrix groups in Las Vegas polynomial time
SODA '92 Proceedings of the third annual ACM-SIAM symposium on Discrete algorithms
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
Computing homomorphisms between holonomic D-modules
Journal of Symbolic Computation - Effective methods in rings of differential operators
Efficient decomposition of separable algebras
Journal of Symbolic Computation
Approximate radical of ideals with clusters of roots
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Deterministic Polynomial Time Algorithms for Matrix Completion Problems
SIAM Journal on Computing
Computation of locally free class groups
ANTS'06 Proceedings of the 7th international conference on Algorithmic Number Theory
The complexity of black-box ring problems
COCOON'06 Proceedings of the 12th annual international conference on Computing and Combinatorics
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The first structure theory in abstract algebra was that of finite dimensional Lie algebras (Cartan-Killing), followed by the structure theory of associative algebras (Wedderburn-Artin). These theories determine, in a non-constructive way, the basic building blocks of the respective algebras (the radical and the simple components of the factor by the radical). In view of the extensive computations done in such algebras, it seems important to design efficient algorithms to find these building blocks.We find polynomial time solutions to a substantial part of these problems. We restrict our attention to algebras over finite fields and over algebraic number fields. We succeed in determining the radical (the “bad part” of the algebra) in polynomial time, using (in the case of prime characteristic) some new algebraic results developed in this paper. For associative algebras we are able to determine the simple components as well. This latter result generalizes factorization of polynomials over the given field. Correspondingly, our algorithm over finite fields is Las Vegas.Some of the results generalize to fields given by oracles.Some fundamental problems remain open. An example: decide whether or not a given rational algebra is a noncommutative field.