A note on the second supplementary law of rational power residue symbols

As a natural generalization of the Legendre symbol, the $q$-th power residue symbol $(a\mathrm{/}p{)}_q$ is defined for primes $p$ and $q$ with $p\equiv 1q$. In this paper, we generalize the second supplementary law by providing an explicit condition for $(q\mathrm{/}p{)}_q=1$, when $p$ has a special form $p={\sum }_{i=0}^{q-1}{m}^i{n}^{q-1-i}$. This condition is expressed in terms of the polylogarithm ${\mathrm{L}\mathrm{i}}_{1-q}(x)$ of negative index. Our proof relies on an argument similar to Lemmermeyer’s proof of Euler’s conjectures for cubic residue.