Show that the polynomial $x^2 + x + 1$ is irreducible in $\mathbb Z/2\mathbb Z[x]$

Question

I've looked up Eisensteins criterion, but I don't understand how to apply it to show that $ x^2 + x + 1$ is irreducible.

Edit: Ok, I see now that Eisensteins criterion does not apply here.

Answer 1

Hint: a quadratic polynomial is reducible over $k$ iff it has a root in $k$.

Answer 2

Since the degree of $f(x) = x^2 + x + 1$ is $\leq 3$, then $f$ is irreducible if and only if $f$ has no roots in $\mathbb{Z}_2$. (why?)

Well, in $\mathbb{Z}_2$, the only possible roots are $0$ and $1$. Try them out!