THE IDEALS OF DEDEKIND 823 classes of algebraic numbers which he called ideals in honor of Kummer's ideal numbers. Before defining Dedekind's ideals let us note the underlying thought. Consider the ordinary integers.In place of the integer 2,Dedekind considers the class of integers 2m,where m is any integer.This class consists of all integers divisible by 2.Likewise 3 is replaced by the class 3n of all integers divisible by 3.The product 6 becom s the collection of all numbers 6p, where p is any integer.Then the product 2.3 =6 is replaced by the state- ment that the class 2m"times"the class 3n equals the class 60.Moreover,the class 2m is a factor of the class 6p,despite the fact that the former class contains the latter.These classes are examples in the ring of ordinary integers of what Dedekind called ideals.To follow Dedekind's work one must accustom oneself to thinking in terms of classes of numbers. More generally,Dedekind defined his ideals as follows:Let K be a specific algebraic number field.A set of integers A of K is said to form an ideal if when a and B are any two integers in the set,the integers uc +v8, where and v are any other algebraic integers in K,also belong to the set Alternatively an ideal d is said to be generated by the algebraic integers c,a,.,n of K if A consists of all sums 入1a1+入2十··+入nan, where the A are e any integers of the field K.This ideal is denoted by (1,aa,.,n).The zero ideal consists of the number 0 alone and accordingly is denoted by(0).The unit ideal is that generated by the number 1,that is, (1).An ideal A is called principal if it is generated by the single integer a, so that (a)consists of all the algebraic integers divisible by a.In the ring of the ordinary integers every ideal is a principal ideal. An example of an ideal in the algebraic number field a+bv-5. where a and b are ordinary rational numbers,is the ideal generated by the integers 2 and 1+-5.This ideal consists of all integers of the form 2u+(1+v-5)v,where u and v are arbitrary integers of the field.The ideal also happens to be a principal ideal because it is generated by the number 2 alone in view of the fact that (1 +V-5)2 must also belong to the ideal generated by 2. Two ideals (a1,a2, .,ap)and (B1,Ba,.,Ba)are equal if every member of the former ideal is a member of the latter and conversely.To tackle the problem of factorization we must first consider the product of two ideals.The product of theideal =(1,.,)and the ideal B=(81,.,B:) of K is defined to be the ideal AB=（a1B1、a1B2,2B1,.,B,.,CBe) It is almost evident that this product is commutative and associative.With this definition we may say that A divides B if there exists an ideal C such that