I am trying to prove that $x^2+1$ is irreducible over $\mathbb{Z}\_p[x]$. In order to do that, let's say it's not irreducible, then we have:

$x^2+1=(x+a)(x+b) \rightarrow x^2+(a+b)x+ab=x^2+1\mod p$, thus we have:

$a=-b \mod p$, $ab=1\mod p \rightarrow b^2=-1\mod p$

Now, can I say that the last equation doesn't have any answer in $\mathbb{Z}[x]\_p$? If yes, how? Because the last equation implies that $b^2 = p-1\mod p$, and I am not sure how to prove there is no $b$ exist to satisfy this equation