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

TL;DR

This paper explores Bernstein's early formulation of Artin's reciprocity law using power residue symbols, highlighting its precedence and overlooked significance before Artin's conjecture.

Contribution

It reveals Bernstein's 1904 formulation of reciprocity law, predating Artin's conjecture, and clarifies its connection to Kummer extensions and roots of unity.

Findings

Bernstein's reciprocity law predates Artin's conjecture.

The law is expressed via power residue symbols.

Bernstein's work was overlooked for nearly 20 years.

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.