Given a field $F$ of 5 elements, show that $F[x]/\langle x^{3}+x+1 \rangle$ is not a field?
I went as follows: We can consider $F$ to be $\mathbb Z_{5}$, so $f(x)=x^{3}+x+1 $ is irreducible in $F$, then $\langle f(x) \rangle$ is maximal, so $F[x]/\langle f(x)\rangle$ is a field!! Where is my mistake??