We present new, efficient algorithms for some fundamental computations with finite-
dimensional(but not necessarily commutative) associative algebras. For a semisimple
associative algebra~ over a finite field or number field F, we show how to compute a basis
for the centre of X and the complete Wedderburn decomposition of 2! as a direct sum of
simple algebras. If~ is given by a generating set of matrices in F~ xm then our algorithm
requires about 0 (rn3) operations in F, plus the cost of factoring a polynomial in F [z] of …

We present new, efficient algorithms for some fundamental computations with finite-
dimensional (but not necessarily commutative) associative algebras over finite fields. For a
semisimple algebra A we show how to compute a complete Wedderburn decomposition of A
as a direct sum of simple algebras, an isomorphism between each simple component and a
full matrix algebra, and a basis for the centre of A. If A is given by a generating set of
matrices inFm× m, then our algorithm requires aboutO (m3) operations inF, in addition to the …