Exercise 29, § 1

a) Let $p$ be a prime number; there exist two integers $a, b$ such that $a^2 + b^2 + 1 \equiv 0 \pmod{p}$ (Alg., Ch. V, 1st edn., §11, No. 5, Cor. of Th. 3). Show that there exist integers $x_1, x_2, x_3, x_4$ not all zero such that $ax_1 + bx_2 \equiv x_3 \pmod{p}$, $bx_1 - ax_2 \equiv x_4 \pmod{p}$ and

(same method as in Exer. 28 c)). Show that $y$ is divisible by $p$, and deduce therefrom that $y = p$.

b) Deduce from a) that every integer $n \geq 0$ is the sum of four squares (Lagrange’s theorem; make use of Alg., Ch. IV, 1st edn., §2, Exer. 11).