Past exam of the options I course of the University of Oxford / 2005 / C / 1 / a

The Legendre symbol isThe definition depends only on the residue class of $a$ modulo the odd prime $p$.

Euler's criterion isIt applies also when $p\mid a$. In particular, for a nonzero residue the power is one or minus one according as the residue is a square or a nonsquare.

For distinct odd primes $p,q$, quadratic reciprocity statesThus the two symbols agree unless both primes are three modulo four, in which case they have opposite signs.