Non-Unique factorizations of algebraic integers

Abstract

We present the main results of the theory of non-unique factorizations as far as they deal with algebraic integers. We specify the philosophy that the class group of an algebraic number field measures to what extent its ring of integers fails to have unique factorization. On the other hand, if the ring of integers fails to have unique factorization then (in a sense to be made precise) almost all integers have many distinct factorizations, but also almost all integers have a clearly arranged set of factorizations.