How to factorize the polynomial in the ring $\mathbb{Z}_5[x]$ I need to factor the polynomial $x^5+3x^4+x^3+x^2+3$ into the product of the irreducible ones in the ring $\mathbb{Z}_5[x]$.
The problem is I don't see any whole roots (I tried every possible divisor of 3) in the given polynomial. Does it mean that it is already irreducible?

It is easy to check that $-1$ is a root, and by Horner's algorithm, you obtain 
$$x^5+3x^4+x^3+x^2+3=(x+1)(x^4+2x^3-x^2+2x-2).$$
Similarly, the quotient has  $2$ as a root and
$$ x^4+2x^3-x^2+2x-2=(x-2)(x^3-x^2+2x+1).$$
The latter quotient has  $-2$ as a root, and
$$x^3-x^2+2x+=(x+2)(x^2+2x+3),$$
so that $$x^5+3x^4+x^3+x^2+3=(x+1)(x-2)(x+2)(x^2+2x+3). $$
Now the quadratic factor is irreducible because it has no root. Indeed
$$ x^2+2x+3=(x+1)^2+2,$$
and the non-zero squares in $\mathbf Z/5\mathbf Z$ are $\pm 1$.
