Elements of a finite field

Question

What is the proof that for any given element $c$ of $F_q$, there exist two elements $a$ and $b$ of $F_q$ such that $a^2 + b^2 = c$. i know that $q$ is the characteristic of this field, but i don't see how this leads, in any way, to a solution to the proof. Thanks a lot.

Answer 1

Observe two things:

• The set $S = \{ a^2 \colon a \in F\}$ has at least $(q+1)/2$ elements (note that at must two elements can have the same square and $0$ has a unique square).

• $c= a^2 +b^2$ with $a,b \in F$ is the same as $S \cap (c-S)$ is non-empty.