  We classify all cubic extensions of any field of arbitrary characteristic, up
to isomorphism, via an explicit construction involving three fundamental types
of cubic forms. We deduce a classification of any Galois cubic extension of a
field. The splitting and ramification of places in a separable cubic extension
of any global function field are completely determined, and precise
Riemann-Hurwitz formulae are given. In doing so, we determine the decomposition
of any cubic polynomial over a finite field.
