Exercise 29, § 1

a) Let $p$ be a prime number; there exist two integers $a, b$ such that $a^2 + b^2 + 1 \equiv 0$ (mod. $p$) (Alg., chap. V, § 11, No. 5, cor. of the theorem 3). Now prove that there exist integers $x_1, x_2, x_3, x_4$ not all zero such that $ax_1 + bx_2 \equiv x_3$ (mod. $p$), $bx_1 - ax_2 \equiv x_4$ (mod. $p$) and

(same method as in Exercise 28 c)). Now prove that $y$ is divisible by $p$, and deduce that $y = p$.

b) Deduce from a) that every integer $n \geq 0$ is the sum of at most four squares (Lagrange's theorem) ; use Alg., chap. IV, § 2, Exercise 11).