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

Addition preserves the condition becauseFor multiplication, write $c_n={\sum }_{i+j=n}{a}_i{b}_j$. Given $q$, choose $N$ so that $a_i,b_i\in p^q{\mathbb{Z}}_p$ for $i\ge N$. If $n\ge 2N$, every pair $i+j=n$ has $i\ge N$ or $j\ge N$, so every summand lies in $p^q{\mathbb{Z}}_p$. Hence $\mathrm{ord}(c_n)\to \infty$, proving that ${\mathbb{Z}}_p⟨T⟩$ is a subring of the formal power series ring ${\mathbb{Z}}_p[[T]]$.

The $(p)$-adic completion isA compatible system of polynomials determines coefficients $a_n\in {\mathbb{Z}}_p$. For each $q$, its reduction has finite degree, so all but finitely many $a_n$ lie in $p^q{\mathbb{Z}}_p$. This is exactly $\mathrm{ord}(a_n)\to \infty$. Conversely, every such restricted series reduces modulo $p^q$ to a polynomial and hence defines a compatible system. Therefore