Past exam of the options II course of the University of Oxford / 2011 / C / C2 / a

For an odd prime number $p$, a nonzero class $a(\mathrm{mod}p)$ is a quadratic residue if $a\equiv x^2(\mathrm{mod}p)$ for some integer $x$ not divisible by $p$. The Legendre symbol isWhen discussing solvability of $x^2\equiv a(\mathrm{mod}p)$ without a coprimality assumption, zero is also a square. This convention matters in the final local characterization of integer squares.

The Euler criterion states that for an odd prime number $p$ and $p\nmid a$,The same modular congruence remains true when $p\mid a$, with both sides zero.

For distinct odd primes $p$ and $q$, quadratic reciprocity statesThe supplementary laws are

If $p$ divides $a$ or $b$, both sides of the claimed multiplicativity identity are zero. Otherwise the Euler criterion givesEach side is $1$ or $-1$. Since $p$ is odd, these two integers represent different residue classes. Hence the modular congruence gives equality: