Past exam of the mathematics course of the University of Cambridge / 2017 / ia / Paper 4 / 5D / b

Suppose $X^2\equiv -1(\mathrm{mod}p)$. Then $X$ is a unit modulo n for modulus $p$, and Fermat's little theorem impliesSince $p$ is an odd prime number, $1$ and $-1$ are distinct residue classes. Therefore $(p-1)/2$ is even, and $p\equiv 1(\mathrm{mod}4)$.

Conversely, put $q=(p-1)/2$. Pair each integer $j\in \{1,\dots ,q\}$ with $p-j$ in the factorial. Wilson theorem yieldsWhen $p\equiv 1(\mathrm{mod}4)$, $q$ is even, so $X=q!$ satisfies $X^2\equiv -1(\mathrm{mod}p)$. This proves the first supplementary law for quadratic reciprocity without assuming a primitive root. HenceThe two solutions are $q!$ and $-q!$ modulo $p$: their difference is nonzero, and factoring $X^2-Y^2$ shows that a degree-two polynomial over the field of residues modulo $p$ has no further roots.