  We establish that any finite extension of function fields of genus greater
than 1 whose relative class group is trivial is Galois and cyclic. This depends
on a result from a preceding paper which establishes a finite list of possible
Weil polynomials for both fields. Given this list, we analyze most cases by
computing options for the splittings of low-degree places in the extension,
then consider the effect of these options on the Weil polynomials of certain
isogeny factors of the Jacobian of the Galois closure. In one case, we use
instead an analysis based on principal polarizations, modeled on an argument of
Howe.
