I have recently been learning about cubic characters, and the machinery of Gauss and Jacobi sums used to prove the cubic reciprocity theorem, and using this, I can now determine when any prime is a cubic residue modulo another prime in terms of the decomposition $4p=L^2+27M^2$.
Any prime except for $3$. I know from various sources that $3$ is a cubic residue modulo $p$ if $3|M$ in the above decomposition. Is there a way to prove this using Gauss and Jacobi sums? All the techniques I have learned that work for every other prime fail for 3.
