On Dedekind's prime ideals

Question

Prime ideals were an essential tool for Dedekind to save or restore unique factorization. Is it fair to say that the shift from Kummer's ideal numbers to Dedekind's ideals (with prime ideals, and so on) corresponds to a shift towards a class of really algebraic numbers (the former being just a precedent, so to speak?

Thanks in advance.

Answer 1

See :

• Richard Dedekind, Theory of Algebraic Integers (French ed 1877 - Translated and introduced by John Stillwell, 1996); you can see Stillwell's introduction, page 3-on :

Dedekind's invention of ideals in the 1870s was a major turning point in the development of algebra. His aim was to apply ideals to number theory, but to do this he had to build the whole framework of commutative algebra: fields, rings, modules and vector spaces. These concepts, together with groups, were to form the core of the future abstract algebra. At the same time, he created algebraic number theory, which became the temporary home of algebra while its core concepts were growing up.

The algebraic integers in Dedekind's title are a generalisation of the ordinary integers - created in response to certain limitations of classical number theory.