DECOMPOSITION OF ALGEBRAS OVER F(Q)(X(1),...,X(M))

Let A be a finite dimensional associative algebra over the field F where F is a finite (algebraic) extension of the function field F(q)(X1,..., X(m)). Here F(q) denotes the finite field of q elements (q = p(l) for a prime p). We address the problem of computing the Jacobson radical Rad (A) of A and the problem of computing the minimal ideals of the radical-free part (Wedderburn decomposition). The algebra sl is given by structure constants over F and F is given by structure constants over F(q) (X1,..., X(m)). We give algorithms to find these structural components of A. Our methods run in polynomial time if m is constant, in particular in the case m = 1. The radical algorithm is deterministic. Our method for computing the Wedderburn deomposition of sl uses randomization (for factoring univariate polynomials over F(q)).