Roots in $\mathbb{Q}_{p}$

Question

Let $\mathbb{Q}_{p}$ denote the set of $p$-adic numbers. For which primes $p\in\{2,3,5,7,11,13,17,19\}$ does the $f(x)=x^{2}+1$ have a root in $\mathbb{Q}_{p}$?

This problem doesn't feel too difficult but I don't know how to start.

Answer 1

For all primes apart from 2 you can use Hensel's lifting lemma.

Answer 2

You’re asking when $\Bbb Q_p$ contains $i=\sqrt{-1}$. The answer is, when $p\equiv1\pmod4$. There are many ways to get to this answer, but perhaps the best is by quadratic reciprocity plus Hensel’s Lemma. QR tells you that $-1$ is a square modulo any $p>2$ under those same conditions on $p$, and Hensel lets you take the relatively prime (in $\Bbb F_p[X]$) factors $X-\tilde i$ and $X+\tilde i$ and lift them to factors of $X^2+1$.