Past exam of the options II course of the University of Oxford / 2013 / C2 / a

For an odd prime $p$, the Legendre symbol is zero if $p\mid a$, one if $a$ is a nonzero quadratic residue modulo $p$, and minus one if it is a quadratic nonresidue. ThusIt depends only on the congruence class of $a$ modulo $p$.

The first supplementary law for quadratic reciprocity isIt also follows immediately from Euler's criterion applied to $-1$.

The second supplementary law for quadratic reciprocity isThe exponent is an integer because every odd square is one modulo eight.

Multiplicativity of the Legendre symbol givesThis includes the case where $p$ divides a factor, when both sides are zero.

For distinct odd primes $p,q$, the quadratic reciprocity law isThus the two symbols have opposite signs exactly when both primes are three modulo four; otherwise they agree. Distinctness excludes zero symbols.