I do not have an answer to Dave Donaghy's question:
> I know this isn't really a GAP question, but does anyone out there know
> if there's a package available that will calculate the invariants of
> the action of a (small) finite group on a purely transcendental field
> extension over some (arbitrary but infinite) base field? The action can be
> given explicitly as the action on the transcendental generators, but ISN'T
> a permutation action. (Even though the group is often the symmetric group.)

but the recommendation to contact Larry Grove (Tucson)
(grove@math.arizona.edu) who has used GAP and Maple in the
investigation of invariants during a visit to Aachen and might have
some advice.

Since this is an area that we have been asked about more often, I
would welcome further comments in the GAP-forum, if somebody has some
hints.

Joachim Neubueser