Past exam of the options II course of the University of Oxford / 2011 / C / C2 / b / ii / Solution

If $\gcd (b,c)>1$, both Jacobi symbols vanish, so the formula holds. Otherwise write $b={\prod }_i{p}_i$ and $c={\prod }_j{q}_j$, retaining repeated factors. Multiplicativity of the Jacobi symbol and quadratic reciprocity giveThe exponent isby the parity identity from part (i). The product of the reversed Legendre symbols is $(c/b)$. Thus the Jacobi reciprocity law isThis also covers $b=1$ or $c=1$ by empty products.