Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 101 / 2 / a / Solution

The ring of p-adic integers is the inverse limitEquivalently, each element has a unique convergent expansion ${\sum }_{j\ge 0}{b}_j{p}^j$ with digits $0\le b_j<p$. It is a complete discrete valuation ring with maximal ideal $(p)$ and residue field ${\mathbb{F}}_p$.