Irreducible polynomial in $\mathbb{Z_2}$

Question

I know that $x^2 + 1$ is irreducible over $\mathbb{R}$, since the $\sqrt{-1}$ is not in the reals.

How can I verify whether or not $x^2 + 1$ is irreducible over $\mathbb{Z_2}$?

So far I have considered that it suffices to show that the polynomial has no reducible factors of degree $1$ or $2$ if it is irreducible.

Answer 1

Since it is quadratic, the only way it could be reducible would be for it to factor completely. So, to show it's irreducible, you only need to show that it has no roots in $\mathbb{Z}/2$. If you find a root, then it is reducible.

Answer 2

Write $Z_2 = \lbrace 0,1 \rbrace$.Then $0^2 + 1 = 1$ and $1^2 + 1 =2 = 0$ in $Z_2$ i.e. $1$ is a root of $X^2 + 1$.Hence it is reducible in $Z_2$.