For a field k, let G be a reductive k-group and V an affine k-variety on which
G acts. Using the notion of cocharacter-closed G(k)-orbits in V , we prove a rational version
of the celebrated Hilbert-Mumford Theorem from geometric invariant theory. We initiate
a study of applications stemming from this rationality tool. A number of examples are
discussed to illustrate the concept of cocharacter-closure and to highlight how it differs from
the usual Zariski-closure.
