p-adic integers

Fix a prime $p$. The ring of $p$-adic integers is the inverse limit

An element is a compatible sequence $(a_n)$, where $a_n$ is a residue class modulo $p^n$ and $a_{n+1}\equiv a_n\pmod{p^n}$. Addition and multiplication are coordinatewise.

Equivalently, $\mathbb Z_p$ is the completion of $\mathbb Z$ for the norm defined by the $p$-adic valuation. Its natural topology is compact, Hausdorff, and totally disconnected. Under addition it is therefore a locally compact group.