<s>[INST] 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?
 [/INST] You need to make it more aggressively.
Check that $2$, $3$ and $4$ are roots, which gives the factorization: 
$$x^5+3x^4+x^3+x^2+3\equiv x^5-2x^4+x^3+x^2-12=$$
$$=x^5-2x^4+x^3-2x^2+3x^2-6x+6x-12=(x-2)(x^4+x^2+3x+6)\equiv$$
$$\equiv(x-2)(x^4-9x^2-2x+6)=(x-2)(x^2(x-3)(x+3)-2(x-3))=$$
$$=(x-2)(x-3)(x^3+3x^2-2)\equiv(x-2)(x-3)(x^3-2x^2-32)=$$
$$=(x-2)(x-3)(x^3-4x^2+2x^2-32)=(x-2)(x-3)(x-4)(x^2+2x+8)=$$
$$=(x-2)(x-3)(x-4)((x+1)^2-3).$$
Easy to see that the quadratic factor is irreducible. 
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