Suppose $\mathbb{Q_p} $ is the fraction field of $\mathbb{Z_p}$ ($p$-adic integers) i.e.
$$\mathbb{Q_p} = \left\lbrace\frac{x}{y} \space \bigg{|} \space x,y \in \mathbb{Z_p} , y\neq 0 \right\rbrace$$
Now with respect to the topology defined by $d(x,y) = e^{-v_p(x-y)}$ ($v_p$ is the $p$-adic valuation) , we need to show that $\mathbb{Q_p} $ is locally compact.
Any suggestions?