What are the ideals of $F_2[x]/\langle x^2 + x +1\rangle$?

Question

Is it just the divisors of $x^2 +x+1$ in mod $2$ ?

Answer 1

Since $\;x^2+x+1\in\Bbb F_2[x]\;$ is irreducible, the ideal $\;I:=\langle x^2+x+1 \rangle\;$ is prime and thus maximal, so $\;\Bbb F_2[x]/I\;$ is a field, and its ideals are...

Answer 2

$x^2+x+1$ is an irreducible polynomial over $\mathbb{F}_2$, since it has no root in $\mathbb{F}_2$.

It follows that $\mathbb{F}_2[x]/(x^2+x+1)$ is a finite field with four elements.

Answer 3

The ideal you are taking the quotient by is prime (and thus maximal, since $F_2[x]$ is a PID). So the quotient is a field. So it's ideals are..what?