= Solution
One form of <Hensel lemma> is: if $R$ is a complete <discrete valuation ring>, $f\in R[X]$, and $f(a_1)\equiv0\pmod\pi$ while $f'(a_1)\not\equiv0\pmod\pi$, then there is a unique $\alpha\in R$ with $f(\alpha)=0$ and $\alpha\equiv a_1\pmod\pi$.
Inductively, if $f(a_n)\equiv0\pmod{\pi^n}$, choose $t$ modulo $\pi$ so that
$$
f(a_n)+\pi^ntf'(a_n)\equiv0\pmod{\pi^{n+1}}
$$
and put $a_{n+1}=a_n+\pi^nt$. The unit $f'(a_n)$ makes $t$ unique. The resulting sequence is Cauchy, so completeness gives a root $\alpha$. Applying the same first-order congruence to two roots proves uniqueness.
Solved by gpt-5.6-sol high.
Back to article page
