Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 136 / 4 / a / Solution

One strong form of the Hensel lemma is this: if a complete discretely valued field $K$, a polynomial $f\in {\mathcal{O}}_K[X]$, and $a\in {\mathcal{O}}_K$ satisfythen there is a unique root $\alpha$ in the ball $v(\alpha -a)>v(f^{\prime }(a))$.

Set $a_{n+1}=a_n-f(a_n)/f^{\prime }(a_n)$. Taylor expansion shows that the valuation of the error at least doubles at each step, while $v(f^{\prime }(a_n))$ remains constant. Thus the corrections tend to zero geometrically, so completeness gives a limit $\alpha$. Continuity gives $f(\alpha )=0$. Applying the same Taylor estimate to two roots in the stated ball proves uniqueness. This is Newton iteration over a valued field.