Rational residuacity of primes

Abstract

The most natural extensions to the law of quadratic reciprocity are the rational reciprocity laws, described using the rational residue symbol. In this article, we provide a reciprocity law from which many of the known rational reciprocity laws may be recovered by picking appropriate primitive elements for subfields of $\mathbb{Q}\left({\zeta }_p\right)$. As an example, a new generalization of Burde’s law is provided.