Problem 9

Let $F$ be a field. We say that $a\in F$ is a square if and only if there exists a $b\in F$ such that $a=b^{2}$. If $a\in F$ is not a square it is called a non-square.

Now let $p$ be an odd prime. Show that if any non-square in $\mathrm{GF}(p)$ is still a non-square in $\mathrm{GF}(p^{n})$, then $n$ is an odd positive integer.(Hint: consider the polynomial $f(x)=x^{2}-a$, where $a$ is a non-square.)