  Let C : y^2=f(x) be a hyperelliptic curve defined over the rationals. Let K
be a number field and suppose f factors over K as a product of irreducible
polynomials f=f_1 f_2...f_r. We shall define a "Selmer set" corresponding to
this factorization with the property that if it is empty then the curve C has
no rational points. We shall demonstrate the effectiveness of our new method by
solving the generalized Fermat equation with signature (3,4,5), which is
unassailable via the previously existing methods.
