I need to show that $\mathbb{Z}_2[X]/(X^2+X+1) = \{0, 1, X^2,X+1 \}$.
My own attempt:
Take $f \in \mathbb{Z}_2[X]$. Then dividing by $X^2+X+1$ gives $f = q(X^2+X+1)+r$ for some $q,r\in\mathbb{Z}_2[X]$ and $\deg(r)<2$. Since this implies that $r$ must be a polynomial in $\mathbb{Z}_2[X]$ of degree $-\infty$, $0$ or $1$, I'd say the only possibilities for $r$ should be $0$, $1$, $X$ and $X+1$.
Is, in this field, $X^2 = X$? If so, how do I see this?
You have $X^2=X^2+X+1+X+1$ and thus $X^2\equiv X+1\pmod {X^2+X+1}$ and thus $$\mathbb{Z}_2[X]/(X^2+X+1)=\{0,1,X,X+1\}$$