The Hasse-Minkowski theorem is the classic local-global principle. It states that a quadratic form has a non-trivial zero globally if and only if it has non-trivial zeroes everywhere locally.

Although the ability to reduce the problem of finding a solution over a single field (Q) to the problem of finding solutions over infinitely many fields (Q_p) does not on the face of it seem particularly useful, one can often use the structure of p-adic fields in order to give short proofs of theorems whose classical proofs can be long and tedius.

To illustrate this power, I will introduce a bit of modern quadratic form theory (i.e. Quadratic Spaces) and then give very elegant proofs of the Three and Four Squares Theorems.