Define the map - : $\mathbb{Z}_{p^{s}}\rightarrow \mathbb{F}_{p}$. Let g(x) be a monic polynomial over $\mathbb{F}_{p}$. A monic polynomial f(x) over $\mathbb{Z}_{p^{s}}[x]$ is called a Hensel lift of g(x) if $\overline{f}(x)=g(x)$ and there is a positive integer n not divisible by p such that $f(x)|(x^{n}-1)$ in $\mathbb{Z}_{p^{s}}[x]$. Otherwise, a monic polynomial g(x) over $\mathbb{F}_{p}$ has a Hensel lift f(x) over $\mathbb{Z}_{p^{s}}[x]$ if and only if g(x) has no multiple roots and $x\nmid g(x)$ in $\mathbb{F}_{p}[x]$.
Could you help me to find Hensel lift f(x) over $\mathbb{Z}_{8}$, if i have a monic polynomial g(x) has no multiple roots and $x\nmid g(x)$ in $\mathbb{F}_{2}[x]$?
For example, $g(x)=x^{3}+x+1\in\mathbb{F}_{2}[x]$.