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

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.

Euler's criterion implies multiplicativity of the symbol on nonzero residues and gives $(-1/p)=(-1{)}^{(p-1)/2}$. Apply quadratic reciprocity with the odd prime 3:The only nonzero square modulo three is one. Since $p>3$,