Notes on Chapter 2 of Dedekind’s Theory of Algebraic Integers

The ring Z consists of the integers of the field Q, and Dedekind takes the theory of unique factorization in Z to be clear and well understood. The problem is that unique factorization can fail when one considers the integers in a finite extension of the rationals, Q(α). Kummer showed that when Q(α) is a cyclotomic extension (i.e. α is a primitive pth root of unity for a prime number p), one can restore unique factorization by introducing “ideal divisors.” Dedekind’s goal is both to improve Kummer’s theory and to extend it to arbitrary Q(α). In Section 5, Dedekind summarizes the properties of the rational integers (i.e. Z) that he would like to extend. In Section 6, Dedekind discusses the Gaussian integers, recalling in particular the notion of the norm, N(ω), of a Gaussian integer ω, and the role of the norm function in showing that Z[i] is a unique factorization domain.