Past exam of the options II course of the University of Oxford / 2004 / C / C2 / a

The Legendre symbol isThe distinction between the zero class and a nonzero quadratic residue is essential.

Euler's criterion statesFor $p\nmid a$ the right side is $+1$ or $-1$. The displayed congruence also holds for $p\mid a$, since $p$ is odd and the exponent is positive.

For distinct odd primes,Equivalently the two Legendre symbols agree unless both primes are three modulo four, in which case their signs are opposite. Neither symbol is zero here.