Topological generator of the additive group of p-adic integers ($\overline{\mathbb{Z}\alpha }={\mathbb{Z}}_p$)

An element $\alpha \in {\mathbb{Z}}_p$ topologically generates the additive group of p-adic integers exactly when $\alpha \not\equiv 0(\mathrm{mod}p)$, equivalently when it is a p-adic unit. Topological generation means that the closure of the cyclic subgroup $\mathbb{Z}\alpha$ is all of ${\mathbb{Z}}_p$.

For a p-adic unit, multiplication by $\alpha$ permutes every finite quotient group $\mathbb{Z}/p^n\mathbb{Z}$. Thus its integer multiples meet every coset $a+p^n{\mathbb{Z}}_p$. These cosets form a neighbourhood basis, so the multiples are dense.

If instead $p\mid \alpha$, every multiple lies in the proper closed subgroup $p{\mathbb{Z}}_p$, making density impossible. In particular, one topologically generates ${\mathbb{Z}}_p$, whereas $p$ does not. The generated algebraic subgroup can remain countable; it is its closure that fills the uncountable topological group.