Simple-root Hensel lifting by successive residues
= Simple-root Hensel lifting by successive residues
For a <complete discrete valuation ring> $R$, let $f(a_1)\in(\pi)$ and let $f'(a_1)$ be a <unit>. If $f(a_n)\in(\pi^n)$, choose $c$ modulo $\pi$ solving $f(a_n)/\pi^n+c f'(a_n)\equiv0$ and put $a_{n+1}=a_n+c\pi^n$. Taylor expansion gives $f(a_{n+1})\in(\pi^{n+1})$. The sequence is <Cauchy> and its limit is a root. If two roots lift the same residue, factoring $f(x)-f(y)$ leaves $x-y$ times a unit, proving uniqueness.