Question 158
Let $n$ and $q$ be two natural numbers, $n \geq 2$, $q \geq 2$ and $q \not \equiv 1(\bmod 4)$, and let $K$ be a finite field with exactly $q$ elements. Prove that for every element $a$ of $K$ there exist $x$ and $y$ in $K$ such that $a = x^{2^{n}} + y^{2^{n}}$. (Every finite field is commutative.)