  We describe an algorithm that determines a set of unramified covers of a
given hyperelliptic curve, with the property that any rational point will lift
to one of the covers. In particular, if the algorithm returns an empty set,
then the hyperelliptic curve has no rational points. This provides a relatively
efficiently tested criterion for solvability of hyperelliptic curves. We also
discuss applications of this algorithm to curves of genus 1 and to curves with
rational points.
