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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, $g(x)=x^{3}+x+1\in\mathbb{F}_{2}[x]$.

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    Maybe before you put up a new problem you should answer the questions that have been piling up in the comments to your previous question.2012-07-21
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    My experience with Galois rings suggests that the OP is looking for a polynomial $f(x)\in \mathbb{Z}_8[x]$ such that $$f(x)\equiv g(x)\pmod 2,$$ and that the order of the coset of $x$ in $\mathbb{Z}_8[x]/\langle f(x)\rangle$ is equal to that of the coset of $x$ in $\mathbb{F}_2[x]/\langle g(x)\rangle$. But I agree with @Gerry. You should try and explain exactly what you want. Even though I like to think I'm good at this kind of guessing games :-)2012-07-29

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