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For which $n\in \{2,3,7\}$ is the polynomial $x^3+x^2+x+2$ irreducible in $\mathbb{Z}/(n)[x]$?

My work:

For a given ring $R$ and a polynomial $p(x)\in R[x]$, if $p(\alpha)=0$ for some $\alpha\in R$, then we can conclude that $p(x)$ is reducible. But if not we cannot say that $p(x)$ is irreducible. So how do we conclude that whether it is irreducible or not in such case? In the question above, if we take $n=3$ or $7$, then we have such situation.

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If our polynomial is of degree $2$ or $3$ over a field, it is irreducible if and only if it has no zero in the field.

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    $\begingroup$ Ok, then what if we have a polynomial of degree greater than $3$. For example $4$? $\endgroup$
    – Extremal
    Jan 1, 2015 at 1:51
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    $\begingroup$ Then things get much harder! $\endgroup$ Jan 1, 2015 at 1:55
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    $\begingroup$ Look at something called the "Sieve Method", it's covered in Artin's book. It helps in seeing if low degree polynomial is irreducible in $\mathbb{F}_{p}$. $\endgroup$
    – user135520
    Jan 1, 2015 at 22:59

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