Dedekind on Higher Congruences and Index Divisors, 1871 and 1878

TL;DR

This paper provides an annotated translation of Dedekind's early works from 1871 and 1878, highlighting their contributions to ideal theory, polynomial congruences, and the problem of common index divisors in algebraic number theory.

Contribution

It offers historical insights and detailed translations of Dedekind's papers, clarifying their role in the development of ideal theory and the understanding of index divisors.

Findings

Dedekind's 1878 paper gives necessary and sufficient conditions for common index divisors.

The work connects ideal theory with polynomial congruences in algebraic number theory.

Historical analysis of Dedekind's original contributions and their influence.

Abstract

Dedekind's theorem connecting ideal theory and polynomial congruences appears in all textbooks on algebraic number theory, but few books note its connection to the problem of ``common index divisors.'' As part of a project to study the history of this problem, we present an annotated translation of two of Dedekind's papers on the subject: a notice about the first publication of Dedekind's ideal theory in 1871 and a paper of 1878 giving proofs of the results announced in 1871 and giving a necessary and sufficient condition for the existence of common index divisors. A separate paper will analyze Hensel's 1894 paper containing the same theorem.