Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 136 / 1 / a / Solution

One form of the Hensel lemma is the following. Let $R$ be a complete discrete valuation ring with maximal ideal $\mathfrak{m}$, and let $f\in R[X]$. If $a_0\in R$ satisfiesthen there is a unique $a\in R$ with $f(a)=0$ and $a\equiv a_0(\mathrm{mod}\mathfrak{m})$.

Define $a_{n+1}=a_n-f(a_n)/f^{\prime }(a_n)$. Since $f^{\prime }(a_n)$ remains a unit, Taylor expansion givesso the valuations of the errors at least double. The corrections tend to zero, making $(a_n)$ a Cauchy sequence; completeness gives a limit $a$, and continuity gives $f(a)=0$. If $a,b$ are two such roots, thenwith $u\in \mathfrak{m}$, so the second factor is a unit and $a=b$.