Past exam of the short options course of the University of Oxford / 2021 / 1 / c / Solution

Write $P={\prod }_i{p}_i$, $Q={\prod }_j{q}_j$, counting repetitions. Since $P,Q$ are coprime, every pair $p_i,q_j$ consists of distinct odd prime numbers. Quadratic reciprocity for the Legendre symbol givesMultiplying over all $i,j$ and using multiplicativity of the Jacobi symbol yields the sign exponentby the odd-product parity identity proved in (b)(i). Thus the Jacobi reciprocity law isIf either integer is $1$, its factorization is empty and both sides equal $1$.