Quadratic character of minus three

For an odd prime $p\ne 3$, quadratic reciprocity givesThus $-3$ is a square modulo $p$ exactly when $p\equiv 1(\mathrm{mod}3)$.

The factorizationshows that $X^3=-1$ always has $X=-1$ as a solution. For $p\ne 3$, it has two further solutions exactly when $-3$ is a quadratic residue; for $p=3$, all three factors coalesce at the unique root $-1$.