Proving that there are infinitely many primes $p$ that $x^2-2$ is irreducible over $\mathbb{Z}_p[x]$ .

Question

$x^2-2$ is not irreducible over $\mathbb{Z}_2[x]$ ($x=\bar{0}$ is a root for $x^2-2$). Similarly for $\mathbb{Z}_3[x]$ ($x^2-2=x^2+2=x^2+3x+2=(x-1)(x-2)$. However $x^2-2$ is irreducible over $\mathbb{Z}_5[x]$. Proving that there are infinitely many $\mathbb{Z}_p[x]$ in which $x^2-2$ is irreducible.

Answer 1

Well, note that you have $\left(\frac{2}{p}\right)=-1$ for primes of the from $\pm{3}\pmod{8}$. And there are infinitely many primes of the form $8n\pm 3$, like $3,5,11,\cdots$

Here $\left(\frac{a}{p}\right)$ denotes the Legendre's Symbol.