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

Write $\chi (a)=(\frac{a}{p})$. When $p\equiv 1(\mathrm{mod}4)$, the first supplementary law for quadratic reciprocity gives $\chi (-1)=1$, so $\chi (p-a)=\chi (a)$. Pairing $a$ with $p-a$ givesThe sum of the nontrivial character $\chi$ over all nonzero residue classes is zero. Because its two half-sums are equal when $\chi (-1)=1$,Hence

By Euler criterion,Choose a primitive root $g$. For any integer $m$ not divisible by $p-1$,For $p>3$, the exponent $m=(p+1)/2$ satisfies $0<m<p-1$. ThereforeTogether with part (i), this is the weighted complete Legendre-symbol sum.