1
$\begingroup$

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.

$\endgroup$
2
$\begingroup$

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.

$\endgroup$
  • 1
    $\begingroup$ Ok, then what if we have a polynomial of degree greater than $3$. For example $4$? $\endgroup$ – Extremal Jan 1 '15 at 1:51
  • 2
    $\begingroup$ Then things get much harder! $\endgroup$ – André Nicolas Jan 1 '15 at 1:55
  • 1
    $\begingroup$ 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 '15 at 22:59

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.