For $\mathbb{F}_p$, I know that the definiton of quadratic residue (and non-residue) is :
$a $ is a quadratic residue if $gcd(a,p) =1$ (i.e $a \neq 0 $ in $\mathbb{F}_p$) and $x^2 \equiv a $ $mod(p)$ has a solution.
$a $ is a quadratic non-residue if $gcd(a,p) =1$ (i.e $a \neq 0 $ in $\mathbb{F}_p$) and $x^2 \equiv a $ $mod(p)$ has no solution.
Now what are the definition in case of $\mathbb{F}_q$ where $q=p^n$? Is the condition $gcd(a,q) =1$ there in the definition?
Also in case of $\mathbb{F}_p$ we get $(p-1)/2 $ residues and $(p-1)/2 $ non-residues. Is there a similar result in case of $\mathbb{F}_q$?