Find all irreducible factors of $x^{10}-1$ in $\mathbb{Z}_3[x]$ I have problem with finding irreducible factors of $x^{10}-1$ in $\mathbb{Z}_3[x]$. Easily one gets that $$x^{10}-1=(x-1)(x+1)(x^8+x^6+x^4+x^2+1)$$ And now I am stuck. I tried dividing the polynomial $x^8+x^6+x^4+x^2+1)$ by irreducible polynomials of order 2 (only three of them but no luck there). Do I have to try now the irreducible polynomials of order 3,4, etc.? Does anyone has simpler/faster/neater solution? Without using Wolfram ;) :) 
Thanks

Certainly one could muse over a faster and neater solution. However, it seems better to apply the Berlekamp algorithm right away, to get
$$
x^8+x^6+x^4+x^2+1=(x^4 + 2x^3 + x^2 + 2x + 1)(x^4 + x^3 + x^2 + x + 1).
$$
And we do not need Wolfram for it, though it is possible to use it.
