Simple algebras are difficult

Let F be a finite field or an algebraic number field. In previous work we have shown how to find the basic building blocks (the radical and the simple components) of a finite dimensional algebra over F in polynomial time (deterministically in characteristic zero and Las Vegas in the finite case). Here we address the more general problem of finding zero divisors in A. This problem is equivalent to finding a nontrivial common invariant subspace of a set of linear operators and includes, as a subcase, the problem of factoring polynomials over the field in question. In [FR] the problem of zero divisors has been reduced, in polynomial time (Las Vegas in the finite case), to the case of simple algebras. We show that, while zero divisors can be found in Las Vegas polynomial time if F is finite, the problem over the rationals might be substantially more difficult. We link the problem to hard number theoretic problems such as quadratic residuosity modulo a composite number. We show that assuming the Generalized Riemann Hypothesis, there exists a randomized polynomial time reduction from quadratic residuosity to determining whether or not a given 4-dimensional algebra over Q has zero divisors. It will follow that finding a pair of zero divisors is at least as hard as factoring squarefree integers. As for the finite case, we give a polynomial time Las Vegas method to construct explicit isomorphisms of matrix algebras. Applications include an algorithm to solve the problem of finding common invariant subspaces for a set of linear operators. Another application answers a question of W. M. Kantor on permutation groups. Finally, as another application of the GRH, we mention a partial result on deterministic factoring over finite fields.