  The Galois group of a parameterized polynomial system of equations encodes
the structure of the solutions. This monodromy group acts on the set of
solutions for a general set of parameters, that is, on the fiber of a
projection from the incidence variety of parameters and solutions onto the
space of parameters. When this projection is decomposable, the Galois group is
imprimitive, and we show that the structure can be exploited for computational
improvements. Furthermore, we develop a new algorithm for solving these systems
based on a suitable trace test. We illustrate our method on examples in
statistics, kinematics, and benchmark problems in computational algebra. In
particular, we resolve a conjecture on the number of solutions of the moment
system associated to a mixture of Gaussian distributions.
