  We classify all cubic function fields over any finite field, particularly
developing a complete Galois theory which includes those cases when the
constant field is missing certain roots of unity. In doing so, we find criteria
which allow one to easily read ramification and splitting data from the
generating equation, in analogy to the known theory for Artin-Schreier and
Kummer extensions. We also describe explicit irreducibility criteria, integral
bases, and Galois actions in terms of canonical generating equations.
