Irreducible quadratic in $\mathbb{Z}_p$

Question

I want to show that for every prime $p$, there exists an irreducible quadratic in $\mathbb{Z}_p[x]$. So I'm looking for some $x^2+ax+b$ that's irreducible. But what $a,b$ choose we choose?

Answer 1

For $p=2$, use $x^2+x+1$. For $p\gt 2$, find a quadratic non-residue $a$ of $p$, there are many, since half of all numbers between $1$ and $p-1$ are quadratic non-residues of $p$. Then $x^2-a$ works.

Answer 2

You don't have to actually find one. It may be easier to count how many reducible quadratics there are and compare it to how many total quadratics there are