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

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


*

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

*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.
