[Math] In $\mathbb{Z}_2[x]$, the ideal $\langle x^2+x+1\rangle$ is maximal

Question

I want to show that in $\mathbb{Z}_2[x]$, the ideal $\langle x^2+x+1\rangle$ is maximal ideal.

I have listed out the table and I can see that the quotient ring is isomorphic to $\mathbb{Z}_2\oplus \mathbb{Z}_2$, but I can't find a ring homomorphism $f:\mathbb{Z}_2[x]\rightarrow \mathbb{Z}_2\oplus \mathbb{Z}_2$ such that the kernel of this homomorphism is exactly the ideal $\langle x^2+x+1\rangle$.

Also, can anyone give me any general suggestion of finding such homomorphisms?

Answer 1

It is a maximal ideal because $x^2+x+1$, a quadratic polynomial with no root in $\mathbf Z_2$, is irreducible in $\mathbf Z_2[x]$ and in PIDs irreducible elements generate maximal ideals.