Suppose I am looking at $GL(4,K)$ acting on a cubic form in say four variables $x,y,z,w$ over $K$ via the usual induced action on a polynomial. Does anyone know what is/where I can find how to compute the ring of invariants? The case of personal interest is when $K$ is a finite field but the the answer over $\mathbb{C}$ would of course be more than useful. 



What is the ring of invariants of GL acting on quaternary cubic forms?
According to Dolgachev in
<http://arxiv.org/abs/math/0408283>
the ring of invariants is generated by 6 invariants of degrees 6, 16, 24, 32, 40 and 100
the last one being a polynomial in the other ones.
In addition to the references to Salmon and Clebsch indicated by Dolgachev, I would also
look up the book by Salmon "A treatise on the analytic geometry of three dimensions"
pp 392-400. It is downloadable from google books:
<http://books.google.com/books?id=LTgDAAAAQAAJ&printsec=frontcover#v=onepage&q&f=false>
The article
<http://arxiv.org/abs/1006.0721>
also contains some references which might help.


