Unique factorization domains

Abstract

Abstract At first it is perhaps rather difficult to appreciate why so much importance is placed on the result that every positive integer (except 1) is a unique product of prime numbers (see 3.14). It is known as the fundamental theorem of arithmetic, which suggests that it must be very significant, but in practice we take it for granted that, for instance, the only way of writing 10 as a product of prime numbers is as 2·5 (or 5 · 2, but that is really the same thing). We might hope that this sort of unique factorization property, when suitably defined, also holds in such systems as the Gaussian integers. If it does, we will have to prove it; we have no right to take it for granted. In fact we shall show that unique factorization does hold for Gaussian integers, but we shall also give examples (one of them apparently rather similar to the Gaussian integers) where unique factorization fails. It is these examples of failure which may bring home to us for the first time the need to be careful. Unique factorization should not be taken for granted, and when it does hold we should appreciate its value in underpinning some of the arithmetic arguments which we may want to use.