Past exam of the options II course of the University of Oxford / 2012 / C / C2 / b / Solution

The quadratic reciprocity states that for distinct odd primes $p,q$,With $q=3$, it gives $\left(\frac{3}{p}\right)=(-1{)}^{(p-1)/2}\left(\frac{p}{3}\right)$. The nonzero square modulo three is one, so $\left(\frac{p}{3}\right)=1$ when $p\equiv 1(\mathrm{mod}3)$, and $-1$ when $p\equiv -1(\mathrm{mod}3)$. An odd prime other than three is congruent to $1,5,7$, or $11$ modulo twelve. Evaluating the two signs givesThus