Past exam of the short options course of the University of Oxford / 2015 / 1 / a

The binomial theorem isolates the only terms that can survive reduction modulo the prime number $p$:For $1\le r\le p-1$, neither $r!$ nor $(p-r)!$ is divisible by $p$, whereas $p!$ is. Consequently $p\mid \left(\genfrac{}{}{0ex}{}{p}{r}\right)$, and every intermediate term vanishes in modular arithmetic. ThusThis works for arbitrary integers $a,b$, including those divisible by $p$. It is the additive identity behind the Frobenius endomorphism in characteristic $p$.

For an odd prime number $p$, Euler's criterion states thatwhere the Legendre symbol is zero if $p\mid a$, one if $a$ is a nonzero quadratic residue, and minus one otherwise. Including the zero case makes this formulation valid for every integer $a$.

Apply Euler's criterion to $a$, $b$, and $ab$. Multiplication in modular arithmetic givesIf $p$ divides either factor, both sides are zero. Otherwise both sides belong to $\{-1,1\}$; these two values are distinct modulo the odd prime number $p$. The congruence is therefore an equality of integers, proving multiplicativity of the Legendre symbol:

For distinct odd prime numbers $p,q$, the law of quadratic reciprocity isThus the two Legendre symbols agree unless both prime numbers are congruent to $3$ modulo $4$, in which case they have opposite signs.