Explain what is meant by a quadratic residue modulo an odd prime $p$, and show that $a$ is a quadratic residue modulo $p$ if and only if $a^{\frac{1}{2}(p-1)} \equiv 1(\bmod p)$. Hence characterize the odd primes $p$ for which $-1$ is a quadratic residue.

State the law of quadratic reciprocity, and use it to determine whether 73 is a quadratic residue (mod 127).