What is the Langlands program?

The aim of this talk is to give a bird's-eye view of the area of mathematics known as the Langlands program.  This is a massive project which attempts to explain those rules by which polynomials factor modulo prime numbers, generalizing the reciprocity laws discovered by Fermat, Euler, and Gauss.  My own work on the subject focuses on the local ("p-adic") situation, so I'll talk about this as well.