How to factor $X^4+5X^3-2X^2-2$ into its irreducible form over $\Bbb{Z}_{11}$

Question

The polynomial $X^4+5X^3-2X^2-2$ has no roots in $\Bbb{Z}_{11}$ so I am unsure as to how I am meant to factorise in such a scenario when I cannot use the factor theorem. How am I meant to progress? And in general for any polynomial with no roots in some finite field $\Bbb{Z}_p$ how should I go about telling if it is already irreducible, and if it isn't how do I know how to factor it?

Answer 1

If $X^4+5X^3-2X^2-2$ has no first degree factors, then it is reducible if and only if it can be written as the product of two quadratic polynomials $X^2+aX+b$ and $X^2+cX+d$. If they exist, then$$\left(X^2+aX+b\right)\left(X^2+cX+d\right)=X^4+5X^3-2X^2-2,$$and therefore $a+c=5$ and $bd=-2=9$. So, for each $b\in\Bbb Z_{11}\setminus\{0\}$, take $d\in\Bbb Z_{11}$ such that $bd=9$, and then see whether or not$$\left(X^2+aX+b\right)\left(X^2+(5-a)X+d\right)=X^4+5X^3-2X^2-2.$$It turns out that, if you pick $b=-1$ and $d=2$, then $a=-2$ will do. That is,$$\left(X^2-2X-1\right)\left(X^2+7x+2\right)=X^4+5X^3-2X^2-2.$$