The l-th Power Reciprocity Law in Regular Cyclotomic Fields

Abstract

The theory of Kummer fields which we have developed thus far gives us the resources required for the proof of certain fundamental laws concerning l-th power residues in regular cyclotomic fields which correspond to the quadratic reciprocity law in the domain of rational numbers and include as a special case the Eisenstein reciprocity law (Theorem 140) between an arbitrary number in k(ζ) and a rational number which we developed in Sect. 115. In order to be able to state these laws for l-th power residues in their simplest form we generalise the symbol {μ/m} defined in Sect. 113 and Sect. 127 as follows.