A complete classification of cubic function fields over any finite field

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.