RO-NMO 2004 Problem 124

Let $\mathcal K$ be a field of characteristic $p$ , $p \equiv 1 \left( \bmod 4 \right)$ .

(a) Prove that $-1$ is the square of an element from $\mathcal K.$

(b) Prove that any element $\neq 0$ from $\mathcal K$ can be written as the sum of three squares, each $\neq 0$ , of elements from $\mathcal K$ .

(c) Can $0$ be written in the same way?

Marian Andronache