Harbingers of Artin's Reciprocity Law. IV. Bernstein's Reciprocity Law

Abstract

In the last article of this series we will first explain how Artin's reciprocity law for unramified abelian extensions can be formulated with the help of power residue symbols, and then show that, in this case, Artin's reciprocity law was already stated by Bernstein in the case where the base field contains the roots of unity necessary for realizing the Hilbert class field as a Kummer extension. Bernstein's article appeared in 1904, almost 20 years before Artin conjectured his version of the reciprocity law, and seems to have been overlooked completely.