Past exam of the mathematics course of the University of Cambridge / 2001 / ia / Paper 4 / 5E (Numbers and Sets)

For a prime number $p$, the nonzero residue classes form the multiplicative group of the finite field ${\mathbb{F}}_p$. Every class has a unique multiplicative inverse. A class equals its own inverse precisely when $x^2=1$, so $(x-1)(x+1)=0$ in the field and $x=1$ or $x=-1$.

For an odd prime, pair each remaining class with its distinct inverse. Each pair contributes one to the product, leavingFor $p=2$, the same congruence is $1\equiv -1(\mathrm{mod}2)$. This proves Wilson's theorem.

Write $r=(p-1)/2$, $O=1\cdot 3\cdots (p-2)$ and $E=2\cdot 4\cdots (p-1)$. The residues $p-2,p-4,\dots ,p-2r$ are exactly the odd factors of $O$, in reverse order. Henceby Wilson's theorem. It follows that the odd-residue squared product modulo a prime isThus it is $-1$ for $p\equiv 1(\mathrm{mod}4)$ and $1$ for $p\equiv 3(\mathrm{mod}4)$.