Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 4 / 11G / a / Solution

For an odd prime $p$, the Legendre symbol isEuler criterion states thatIf $p\mid a$, both sides vanish. Otherwise choose a primitive root $g$ and write $a\equiv g^m$. Since $g^{(p-1)/2}\equiv -1$,The power $g^m$ is a square exactly when $m$ is even, proving the criterion. Taking $a=-1$ gives the first supplementary law for quadratic reciprocity