Prove that $(x^2 + x+2)$ ideal is a maximal ideal in $F_3[x]$ ring.
I know the theorem about factor rings and fields
B ideal from A ring is maximal iff A/B factor-ring is a field
How can I use the theorem here?
If $F$ is a field then a non-zero polynomial $f(x) \in F[x]$ is irreducible if and only if the ideal $(f)$ is prime if and only if the ideal $(f)$ is maximal. All you need to do is show that $x^2 + x + 2$ is irreducible. This is easy because all the possible factors are linear.