Algebraic number theory seeks to understand and somehow classify all Galois extensions of a number field $K$. One perspective on this basic classification problem emerges from the Cebotarev density theorem, which implies that such extensions $L/K$ are determined by the primes of $K$ that split completely in $L$. When $L/K$ is abelian, associating such a 'splitting law' is classical, and after defining the relevant terms I will illustrate this theory with some representative examples. The corresponding question for non-abelian extensions remains a great mystery, however, but I hope to use this to motivate a very concrete introduction to some of the modern machinery of number theory and the Langlands program. No knowledge of number theory will be assumed.