We begin with the definition of the -adic numbers for any positive integer . Section 14.2.1 is about the -adics in the special case ; these are fun because they can be represented as decimal expansions that go off infinitely far to the left. Section 14.2.3 is about how the topology of is nothing like the topology of . Finally, in Section 14.2.4 we state the Hasse-Minkowski theorem, which shows how to use -adic numbers to decide whether or not a quadratic equation in variables has a rational zero.