The invention of ideals by Dedekind in the 1870s was well ahead of its time, and proved to be the genesis of what today we would call algebraic number theory. His memoir 'Sur la Theorie des Nombres Entiers Algebriques' first appeared in instalments in the 'Bulletin des sciences mathematiques' in 1877. This is a translation of that work by John Stillwell, who also adds a detailed introduction that gives the historical background as well as outlining the mathematical obstructions that Dedekind was striving to overcome. Dedekind's memoir gives a candid account of his development of an elegant theory as well as providing blow-by-blow comments as he wrestled with the many difficulties encountered en route. A must for all number theorists.
Table of Contents
Part I. Translator's Introduction: 1. General
3. Quadratic forms
4. Quadratic integers
5. Roots of unity
6. Algebraic integers
7. The reception of ideal theory
Part II. Theory of Algebraic Integers: 8.
Auxiliary theorems from the theory of modules
9. Germ of the theory of ideals
10. General properties of algebraic integers
11. Elements of the theory of ideals.