Cubic Fields: A Primer

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.