Jacobi and Kummer's Ideal Numbers

TL;DR

This paper provides a modern interpretation of Kummer's ideal numbers, tracing their development from Jacobi's cyclotomy work and his methods for analyzing Jacobi sums from the 1830s.

Contribution

It offers a new perspective on Kummer's ideal numbers by connecting them to Jacobi's earlier cyclotomy techniques and Jacobi sums.

Findings

Historical connection between Jacobi's cyclotomy and Kummer's ideal numbers

Clarification of the development of ideal numbers from Jacobi's work

Enhanced understanding of the origins of algebraic number theory

Abstract

In this article we give a modern interpretation of Kummer's ideal numbers and show how they developed from Jacobi's work on cyclotomy, in particular the methods for studying "Jacobi sums" which he presented in his lectures on number theory and cyclotomy in the winter semester 1836/37.