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

Abstract

As a natural generalization of the Legendre symbol, the $q$-th power residue symbol $(a/p{)}_q$ is defined for primes $p$ and $q$ with $p\equiv 1\mathrm{mod}q$. In this paper, we generalize the second supplementary law by providing an explicit condition for $(q/p{)}_q=1$, 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 ${\mathrm{Li}}_{1-q}(x)$ of negative index. Our proof relies on an argument similar to Lemmermeyer's proof of Euler's conjectures for cubic residue.

17 pages. In version 2, the data in Example 1.6 has been corrected