Eecient Decomposition of Separable Algebras

We present new, eecient algorithms for computations on separable matrix algebras over innnite elds. We provide a probabilistic of the Monte Carlo type to nd a generator for the centre of a given algebra A F mm over an innnite eld F using a number of operations that is within a logarithmic factor of the cost of solving m m systems of linear equations. A Las Vegas algorithm is also provided under the assumption that a basis and set of generators for the given algebra are available. These new techniques yield a partial factorization of the minimal polynomial of the generator that is computed, which may reduce the cost of computing simple components of the algebra in some cases.