Is there any method to find out the characterization of all primes $p$ such that $\frac{a}{b}$ is a quadratic residue modulo $p$ such that $a$ and $b$ are primes? Is there any method to do the same for cubic residues?
The quadratic reciprocity laws I know only give me an idea on how to find out whether an integer is a residue or not. I have no idea on how the same could be extended to cubics too.
If there is no characterization for general $a,b$ as primes, can you provide the characterization for all $a,b < 10$? (Both quadratic and cubic). Any help will be appreciated.