$ 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.

William Stein 2004-05-06