Irreducible polynomial over $\mathbb{Z}_p$

Question

Let $p$ be prime. Is there a specific form for $p$ in which

(1) $x^2+1$ is irreducible over $\mathbb{Z}_p$?

(1) $x^2-1$ is irreducible over $\mathbb{Z}_p$?

Thanks!

Answer 1

First note that $x^2-1=(x-1)(x+1)$ is always reducible over $\mathbb{Z}_p$, for any positive prime integer $p$. For the remaining part, $x^2+1=x^2-1$ when $p=2$, and for odd prime integers, see the answer given by mathworker21 of the following posted question Prime integer p such that -1 is is a square mod p. Indeed, $x^2+1$ is reducible over $\mathbb{Z}_p$ if and only if $x^2+1$ has a root in $\mathbb{Z}_p$ if and only if $-1\equiv p-1$ mod $p$ is a perfect square in $\mathbb{Z}_p$.