Square root in $\mathbb{Q}_p$:
Assume $p$ is an odd prime. Consider a p-adic unit $\alpha=a_0+a_1p+a_2p^2+\cdots \in \mathbb{Z}_p$.
The question is- $\text{which such $\alpha$ has square root in $\mathbb{Q}_p$}$ ?
It turns out that-
$\alpha$ has square root in $\mathbb{Q}_p$ iff $a_0$ is quadratic residue modulo $p$.
Now for $p \neq 2$, $-\frac{1}{2}$ is a $p$-adic unit in $\mathbb{Z}_p$. Let us check whether $-1/2$ has a square root in $\mathbb{Q}_3$. For that, $$ -1/2=\frac{1}{1-3}={\color{blue}{1}}+3+3^2+\cdots \ \text{and} \ {\color{blue}{1}}\equiv 2^2 \ \text{mod}\ 3. $$ Thus $-1/2$ has a sqaure root in $\mathbb{Q}_3$.
Is it equivalent to say that $+1/2$ also has square root in $\mathbb{Q}_3$ ?
Similarly, I have checked that $-1/2$ has square roots in $\mathbb{Q}_{11}$, $\mathbb{Q}_{17}$, $\mathbb{Q}_{19}, \cdots $
$(1)$ What are the all primes $p$ so that $-1/2$ is a square in $\mathbb{Q}_p$ ?
$(2)$ what about the general formula so that $\pm \frac{m}{n}$ has square roots in $\mathbb{Q}_p$, $ m,n \in \mathbb{N}$ ?