The development of the principal genus theorem

Abstract

Genus theory today belongs to algebraic number theory and deals with a certain part of the ideal class group of a number field that is more easily accessible than the rest. Historically, the importance of genus theory stems from the fact that it was the essential algebraic ingredient in the derivation of the classical reciprocity laws, from Gauss’s second proof, via Kummer’s contributions, all the way to Takagi’s reciprocity law for p-th power residues.