Question 12
Consider two integers $n$ and $q$, $n \ge 2$, $q \ge 2$, and $q \ne 1 \pmod 4$. Let $K$ be a finite field having exactly $q$ elements. Show that for any $a$ from $K$, there is $x$ and $y$ in $K$, such that $a = x^2 + y^2$. (It is known that any finite field is commutative.)