What are the zero divisors of $\mathbb F_2[x]/\langle x^2+1\rangle$?
I think it is not a field, since the polynomial is reducible over $\mathbb F_2$. And because the ring is finite one has to show that it has a zero divisor, is it:
$$\mathbb F_2[x]/\langle x^2+1\rangle =\mathbb F_2[x]/\langle (x+1)^2\rangle $$
I'm stuck here, does the latter imply that $x^2=x=1$, so are there less than $4$ elements, otherwise I cannot find a zero divisor, can you help ?