Past exam of the probability, statistics and options I course of the University of Oxford / 2010 / C / 1 / iii

Write $s=(a/p)$. If $p\mid a$, both sides of the proposed identity are zero. Otherwise $s=\pm 1$. The multiplicativity of the Legendre symbol and Euler's criterion giveFor the middle equality, Euler's congruence determines the sign exactly, since both the symbol and $s^{(p-1)/2}$ are in $\{1,-1\}$ and $p$ is odd.