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

Any complex root of $x^{10}-1$ is a tenth root of unity, hence
$$ x^{10}-1 = (x-1)(x+1)\Phi_5(x)\Phi_{10}(x) \tag{1}$$
and the splitting field of $\Phi_n(x)$ (that is the minimal polynomial over $\mathbb{Q}$ of a primitive $n$-th root of unity) over $\mathbb{F}_{p}$ is given by $\mathbb{F}_{p^d}$ with $d$ being the least integer such that $n\mid(p^d-1)$. Since $3$ is a generator for both $\mathbb{Z}/(5\mathbb{Z})^*$ and $\mathbb{Z}/(10\mathbb{Z})^*$, $\Phi_5$ and $\Phi_{10}$ are irreducible over $\mathbb{F}_3$ and $(1)$ is a factorization over $\mathbb{F}_3$.
