In this thread on his weblog, Terry Tao mentioned an exciting paper by Noga Alon. The paper explains a result that Alon calls the Combinatorial Nullstellensatz. The ordinary Nullstellensatz relates algebra and geometry over algebraically closed fields. Consider the set of common zeroes of a system of (multivariate) polynomial equations over the field. Then a polynomial vanishes on that set if and only if a power of the polynomial is in the ideal generated by the system of equations.
It is easy to see that no such simple result holds over non-algebraically closed fields. Alon is able to prove an analogue of the Nullstellensatz in a very special case, so special that it is not of particular interest of itself. But he is able to use it to give new short proofs of many existing results, such as the Chevalley-Warning and Cauchy-Davenport theorems, as well as many results in combinatorics.