p-adic integer (${\mathbb{Z}}_p$)

For a prime number $p$, the ring of p-adic integers is the inverse limitEvery p-adic integer has a unique convergent expansion ${\sum }_{j\ge 0}{b}_j{p}^j$ with $0\le b_j<p$. It is a complete discrete valuation ring with uniformizer $p$ and residue field ${\mathbb{F}}_p$.

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 $\mathbb{Z}/p^n\mathbb{Z}$.