Let $F$ be a finite field and $\tau$ an element of $F$. Prove that there exists $a,b\in F$ such that $\tau=a^2+b^2$.

Question

Let $F$ be a finite field and $\tau$ an element of $F$. Prove that there exists $a,b\in F$ such that $\tau=a^2+b^2$. It suffices to prove for the case $F=\mathbb{Z}_p$. How to prove?

Answer 1

Let $|F| = p^n$, where $p$ is a prime; and consider $\varphi : F\to F$ given by $x\mapsto x^2$.

1. If $p=2$, $\varphi$ is an isomorphism, so we're done.

2. If $p > 2$, check that $\varphi(x) = \varphi(y)$ iff $x = \pm y$, and hence (why?) $$ |Im(\varphi)| \geq \frac{p^n+1}{2} := k $$ For $z \in F$, consider $S:= \{z - \varphi(x) : x \in F\}$. Since $2k > p^n$, one has that $S\cap Im(\varphi) \neq \emptyset$, so we are done.