I have seen that you can define the usual modular arithmetic on the $p$-adic integers: For $a,b\in\mathbb{Z}_p$ and a prime $p$, $$a\equiv b\pmod{p}\iff (a-b)/p\in \mathbb{Z}_p.$$
My question is, can we also define the Legendre symbol of $p$-adic integers?
For example, for a prime $p$ and $a\in\mathbb{Z}_p$, define $$\left(\frac{a}{p}\right) = \begin{cases} 1\quad\text{if $x^2\equiv a\pmod{p}$ has a solution $x\in\mathbb{Z}_p$} \\ -1\quad\text{if $x^2\equiv a\pmod{p}$ has no solution $x\in\mathbb{Z}_p$}\\ 0\quad\text{if $a\equiv0\pmod{p}$}\end{cases}.$$
Can I use the usual properties of the Legendre symbol in this case?