Determine if the polynomial $X^4+2X^2+1 \in \Bbb Z_3[X]$ is non divisible.
I couldn’t find any theorems on ”non divisiblity” so I assume that this isn’t divisible since it doesn’t have any roots in $\Bbb Z_3$? We have that $0+0+1=1, 1+2+1=4=1$ and $2^4+8+1=25=1$. Wouldn’t this imply that the polynomial doesn’t factor and thus doesn’t have any divisor?
Over any field you have $X^4+2X^2+1=(X^2+1)^2$, and therefore $X^4+2X^2+1$ is reducible, even when it has no roots. It's only when the degree of polynomial is $2$ or $3$ that the non-existence of roots implies that the polynomial is irreducible.