After the somewhat traumatic realization that factorization into irreducibles is unique in some rings of integers but not in others, we seek some way to minimize the damage. Kummer, and then Dedekind, took steps to develop more insightful theories. Kummer had the bright idea that if he could not factorize a number uniquely in a given ring of integers, then perhaps he could extend the ring to a bigger one in which further factorization might be not only possible, but unique. For example, we pointed out in Chapter 4 that there are two factorizations 6 = 2 · 3 =√−6 · √−6 in Z[√−6], but √−6 does not divide 2 or 3 in this ring. In fact, 2/
√−6 =√−2/3, and 3/√−6 =√−3/2, neither of which belongs to Z[ √−6]. Kummer’s idea: throw them into the pot to create a larger ring.
He called the new elements introduced in this way ‘ideal numbers’. Dedekind looked at the same ideas from a different direction, introduc-
ing the notion of an ‘ideal’ in ring theory: a special kind of subring. The word referred to the reformulation of Kummer’s notion of ideal numbers. Dedekind showed that although unique factorization may fail for numbers, a simple and elegant theory of unique factorization can be developed for ideals. In this theory, the essential building blocks are ‘prime ideals’, which are defined by adapting the definition of a prime element from the previous chapter.