Re: factorization in Z_3

In article <1118456591.708641.188280@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> "Li Yi" <liyi.cn@xxxxxxxxx> writes:
> Factor x^9-x and x^{27}-x in Z_3. Prove that your factorization are
> irreducible.
>
> I've really have no idea about this.
> For the first one, we need to factorize x^4+1 in Z_3.
> How to do that?
>
> And for the second one, we have to factorize a polynomial of degree 12!
>
> Can anyone give some hints?

You have already found that 0 and 1 are roots. You might start to check
whether there are other roots in Z_3. Next to factorise x^4 + 1 in Z_3
you may note that when it factors it will either factor as two quadratics,
or as a linear term and a cubic. It is easy to verify that it does not
have a linear factor, so you may try two quadradics. There are only
9 essentially different in Z_3, and a lot of them you can discard
immediately.
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