We will present a classification of all cubic extensions of any field of arbitrary characteristic, up to isomorphism, via an explicit construction involving three fundamental types of one parameter cubic forms. We will also classify any Galois cubic extension of a field. The splitting and ramification of places in a separable cubic extension of any global function field will be completely determined, and precise Riemann-Hurwitz formulae will be given. In doing so, we will determine the decomposition of any cubic polynomial over a finite field.
This is joint work with Kenneth Ward.