Finding squares in finite fields

Question

I need the definition of finding squares in finite fields and also the number of squares in a finite field. How can we find squares in $\Bbb F_5$ and $\Bbb F_7$? (Here $\Bbb F_5$ and $\Bbb F_7$ indicate the finite fields for $q=5$ and $q=7$ respectively.)

Can we generalize for different values of $q$? I need your help.

Thank you.

Answer 1

Hint: In a field with characteristic different from $2$, if $a\ne 0$ the equation $x^2=a$ has $0$ or $2$ solutions.

If you are interested in finding the squares in a field with $5$ elements, or one with $7$, you know what thse fields look like, square each element and list the answers you get.

Answer 2

If $F_n$ is a finite field where $n$ is an odd prime, then the perfect squares in $F_n$ are namely the quadratic residues $\pmod n$. For example, in $F_5$, $1$ and $4$ are the only non-zero elements which are perfect squares, because $1^2 = 1, 2^2 = 4, 3^2 = 4, 4^2 = 1$. Answer made simple.