Find the prime factorization of $x^3-x+1$ in $\Bbb Z_3[x]$ and in $\Bbb Z_5[x]$.

Question

Find the prime factorization of $x^3-x+1$ in $\Bbb Z_3[x]$ and in $\Bbb Z_5[x]$.

The polynomial $x^3-x+1$ no roots in $\Bbb Z_3[x]$ and is of degree $3$ so I think it implies that it's irreducible and has no factorization?

Similarly it has not roots in $\Bbb Z_5[x]$ so it's irreducible. Have I understood something wrong or is the question just poorly worded?

Answer 1

The polynomial $x^3-x+1$ is irreducible in $\Bbb Z_3[x]$ but not in $\Bbb Z_5[x]$. Notice that $$(-2)^3-(-2)+1=-8+2+1=-5\equiv 0 \pmod{5}.$$ After the division by $(x+2)$, we find that its factorization in $\Bbb Z_5[x]$ is $$x^3-x+1=(x+2)(x^2+3x+3).$$ It is easy to verify that $x^2+3x+3$ is irreducible in $\Bbb Z_5[x]$.