Q_p is locally compact and totally disconnected

By [L1], ${\mathbb{Z}}_p$ is the closed unit ball, and by the clopen-ball lemma it is open. To prove compactness, note first that ${\mathbb{Z}}_p$ is complete as a closed subset of the complete field ${\mathbb{Q}}_p$. It is totally bounded because for each $N\ge 1$, every element of ${\mathbb{Z}}_p$ differs by at most $p^{-N}$ from one of the finitely many truncations $a_0+a_{1}p+\cdots +a_{N-1}{p}^{N-1}$ with digits $a_i\in \{0,\dots ,p-1\}$, by [L2]. Hence [L3] makes ${\mathbb{Z}}_p$ compact.

Every point of ${\mathbb{Q}}_p$ has a compact open neighborhood, namely a scalar multiple of ${\mathbb{Z}}_p$, so ${\mathbb{Q}}_p$ is locally compact. If $x\mathrm{\ne }y$, choose a ball around $x$ whose radius is smaller than $\mathrm{\mid }x-y{\mathrm{\mid }}_p$; by [L1] this ball is clopen and does not contain $y$. Therefore points are separated by clopen sets, so ${\mathbb{Q}}_p$ is totally disconnected.