In general, to show that $a$ is quadratic residue modulo $n$? What do I have to show? I'm always struggling with proving a number $a$ is quadratic residue or non-quadratic residue.
For example,
If $n = 2^{\alpha}m$, where $m$ is odd, and $(a, n) = 1$. Prove that $a$ is quadratic residue modulo $n$ iff the following are satisfied:
If $\alpha = 2$ then $a \equiv 1 \pmod{4}$.
I just want to know what do I need to show in general, because I want to solve this problem on my own. Any suggestion would be greatly appreciated.
Thank you.