$p$-adic Numbers

This section is about the $p$-adic numbers $\mathbf{Q}_p$, which are the completion of $\mathbf{Q}$ with respect to the $p$-adic valuation. Alternatively, to give a $p$-adic integer in $\mathbf{Z}_p$ is the same as giving for every prime power $p^r$ an element $a_r\in \mathbf{Z}/p^r\mathbf{Z}$ such that if $s\leq r$ then $a_s$ is the reduction of $a_r$ modulo $p^s$. The field $\mathbf{Q}_p$ is then the field of fractions of $\mathbf{Z}_p$.

We begin with the definition of the $N$-adic numbers for any positive integer $N$. Section 16.2.1 is about the $N$-adics in the special case $N=10$; these are fun because they can be represented as decimal expansions that go off infinitely far to the left. Section 16.2.3 is about how the topology of $\mathbf{Q}_N$ is nothing like the topology of $\mathbf{R}$. Finally, in Section 16.2.4 we state the Hasse-Minkowski theorem, which shows how to use $p$-adic numbers to decide whether or not a quadratic equation in $n$ variables has a rational zero.