Kurt Hensel on Common Inessential Discriminant Divisors, 1894

TL;DR

This paper provides an annotated translation of Hensel's 1894 work on algebraic number theory, focusing on the concept of common inessential discriminant divisors, a topic historically studied by Dedekind, Kronecker, and Hensel.

Contribution

It offers an accessible translation and detailed analysis of Hensel's 1894 paper, highlighting its significance in the study of discriminant divisors in algebraic number theory.

Findings

Analysis of common inessential discriminant divisors

Historical context of Hensel's contributions

Clarification of algebraic number theory concepts

Abstract

The problem of the "common inessential discriminant divisors" attracted the attention of Dedekind, Kronecker, and Hensel in the early days of algebraic number theory. Four sources are particularly important: Dedekind's announcement, in 1871, of the second edition of Dirichlet's lectures \cite{anzeige}, Dedekind's 1878 paper, the 25th section of Kronecker's 1882 \textit{Grundz\"uge}, and Hensel's 1894 paper, which is our focus here. Both of the key papers of Dedekind were translated and annotated in our paper "Dedekind on higher congruences and index divisors, 1871 and 1878." (arXiv:2107.08905}. We here present an annotated translation of Kurt Hensel's "Arithmetische Untersuchungen \"uber die gemeinsamen ausserwesentlichen Discriminantentheiler einer Gattung" (\textit{Journal f\"ur die Reine und Angewandte Mathematik}, \textbf{113} (1894), 128--160).