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We present new, efficient algorithms for some fundamental computations with finitedimensional (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 in F m×m , then our algorithm requires about O(m 3 )doi:10.1006/jsco.1999.0308 fatcat:myr6prcksrge3fufgqbxa6rlsa