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LetA be a finite dimensional associative algebra over the fieldF whereF is a finite (algebraic) extension of the function fieldF q(X 1,...,X m). Here Fq denotes the finite field ofq elements (q=pl for a primep). We address the problem of computing the Jacobson radical Rad (A) ofA and the problem of computing the minimal ideals of the radical-free part (Wedderburn decomposition). The algebraA is given by structure constants overF andF is given by structure constants overF q(X 1,...,X m). We give algorithms to find these structural components ofA. Our methods run in polynomial time ifm is constant, in particular in the casem=1. The radical algorithm is deterministic. Our method for computing the Wedderburn decomposition ofA uses randomization (for factoring univariate polynomials overF q).