Cube roots of minus one modulo an odd prime
= Cube roots of minus one modulo an odd prime
The factorization
$$
X^3+1=(X+1)(X^2-X+1)
$$
shows that $X^3=-1$ always has $X=-1$ as a solution. For $p\ne3$, it has two further solutions exactly when $-3$ is a quadratic residue; for $p=3$, all three factors coalesce at the unique root $-1$.