 Topological generator of the additive group of p-adic integers

An element $\alpha \in {\mathbb{Z}}_p$ topologically generates the additive group of ${\mathbb{Z}}_p$ exactly when $\alpha \not\equiv 0(\mathrm{mod}p)$, equivalently when $\alpha$ is a p-adic unit. In that case its image generates every cyclic quotient $\mathbb{Z}/p^n\mathbb{Z}$.