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
824 THE THEORY OF NUMBERS 1800-1000 B AC.One writes A|B and A is called a factor of B.As already suggested above by our example of the ordinary integers,the elements of B are included in the elements of A and ordinary divisibility is replaced by class inclusion. The ideals that are the analogues of the ordinary prime numbers are called prime ideals.Such an ideal P is defined to be one which has no factors othcr than itself and the ideal (1),so that P is not contained in any other ideal of K.For this reason a prime ideal is also called maximal.All of these definitions and theorems lead to the basic theorems for ideals of an algebraic number field K.Any ideal is divisible by only a finite number of ideals and if a prime ideal divides the product AB of two ideals (of the same number class)it divides A or B.Finally the fundamental theorem in the theory of ideals is that every ideal can be factored uniqucly into prime ideals In our earlier examples of algebraic number fields of the form a +bvD, Dintegral,we found that some permitted unique factorization of the algebraic integers of those fields and others did not.The answer to the question of which do or do not is given by the theorem that the factorization of the integers of an algebraic number field K into primes is unique if and only if all the ideals in K are principal. From these examples of Dedekind's work we can see that his theory of ideals is indeed a gencralization of the ordinary integers.In particular,it furnishes the concepts and properties in the domain of algebraic numbers which enable one to establish que factorization Leopold Kronecker (1823-91),who was Kummer's favorite pupil and who succeeded Kummer as professor at the University of Berlin,also took up the subject of algebraic numbers and dcveloped it along lines similar to Dedekind's.Kronecker's doctoral thesis "On Complex Units,"written in 1845 though not published until much later,19 was his first work in the subject.The thesis deals with the units that can exist in the algebraic number fields created by gauss. Kronccker created another theory of fields(domains of rationality).20 His field concept is more gencral than Dedekind's because he considered fields of rational functions in any number of variables (indeterminates) Specifically Kronecker introduced (1881)the notion of an indcterminate adjoined to a ficld,the indeterminate bcing just a new abstract quantity This idea of extending a field by adding an indeterminate he made the cornerstone of his theory of algebraic numbers.Here he used the knowledge that had been built up by Liouville,Cantor,and others on the distinction between algebraic and transcendental nurnbers.In particular he observed that if x is transcendental over a ficld K(x is an indeterminate)then the ficld 19.Jour fur Math,.93,1882,1-52=Werke,1,5-71. 20."Grundzuge ciner arithmetischen Theorie der algebraischen (,rossen,"Jour fir Math.,92,1882,1-122 Werke,2,237-387;also published separately by (Rcimer, 1882
THE IDEALS OF DEDEKIND 825 K(x)obtained by adjoining the indeterminate x to K,that is,the smallest field containing K and x,is isomorphic to the field K[]of rational functions in one variable with coefficients in K.21 He did stress that the in- determinate was merely an element of an algebra and not a variable in the sense of analysis.He then showed in 1887 that to each ordinary prime number there corresponds within the ring Q()of polynomials with rational coefficients a prime polynomial (x)which is irreducible in the rational field Q.By considering two polynomials to be equal if they are congruent modulo a given prime polynomial p(x),the ring of all polynomials in Q(x) becomes a field of residue classes possessing the same algebraic properties as the algebraic number field K(8)arising from the field K by adjoining a root 8 ofp()=0.Here he used the idea Cauchy had already employed to introduce imaginary numbers by using polynomials congruent modulo x2+1.In this same vork he showed that the theory of algebraic numbers is independent of the fundamental theorem of algebra and of the theory of the complete real number system In his theory of fields (in the "Grundzuge")whose elements are formed by starting with a field K and then adjoining indeterminates ,Kronecker introduced the notion of a modular system that played the role of ideals in Dedekind's theory.For Kronecker a modular system is a set M of those polynomials in n variables *1, .x such that if P and P2 belong to the set so does P+P2 and if P belongs so does QP where is any polynomial in 1,x A basis of a modular system M is any set of polynomials B,B2.of M such that every polynomial of M is expressible in the form RB1+R2B2+· where R,R2,.are constants or polynomials (not necessarily belonging to M).The theory of divisibility in Kr cker's general fields was defined in terms of modular systems much as Dedekind had done with ideals The work on algebraic number theory was climaxed in the nineteenth century by Hilbert's famous report on algebraic numbers.24 This report is primarily an account of what had been done during the century.However, Hilbert r worked all of this earlier theory and gave new,elegant,and power- ful methods of securing these results.He had begun to create new ideas in algebraic number theory from about 1892 on and one of the new creations on Galoisian number fields was also incorporated in the report.Subsequently Hilbert and many other men extended algebraic number theory vastly. 21.Wrke.2.253 29.7xk-。2.330 23.Jow.fur Math.100.1B87.490-510=Verke.3.211-40. 24."Die Theorie der algebraischen Zahlkorper"(The Theory of Algebraic Number Fields).Jahres.der Deut.Math.-Ver.,4,1897,175-546=Ges.Abh.1,63-363
826 THE THEORY OF NUMBERS 1800-1900 However,these later developments,relative Galoisian fields,relative Abelian number fields and class fields,all of which stimulated an immense amount of work in the twentieth century,are of concern primarily to specialists. Algebraic number theory,originally a scheme for investigating the solutions of problems in the older theory of numbers,has become an end in itself.It has come to occupy a position in between the theory of numbers and abstract algebra,and now number theory and modern higher algebra merge in algebraic number theory.Of course algebraic number theory has also produced new theorems in the ordinary theory of numbers. 5.The Theory of Forms Another class of problems in the theory of numbers is the representation of integers by forms.The expression ax2 2bxy cy2, wherein a,b,and c are integral,is a binary form because two variables are involved,and it is a quadratic form because it is of the second degree.A number M is said to be represented by the form if for specific integral values of a,b,c,*and y the above expression equals Mf.One problem is to find the set of numbers M that are representable by a given form or class of forms The converse problem,given M and given a,b,and c or some class of a,b, and c,to find the values of x and y that represent M,is equally important. The latter problem belongs to Diophantine analysis and the former may equally well be considered part of the same subject. Euler had obtained some special results on these problems.However, Lagrange made the key discovery that if a number is representable by one form it is also representable by many other forms,which he called equivalent. The latter could be obtained from the original form by a change of variables x=ax+y到 y =yx'+8y' wherein c,B,y,and 8 are integral and a-By =1.35 In particular, Lagrange showed that for a given discriminant (Gauss used the word determinant)62-ac there is a finite number of forms such that each form with that discriminant is equivalent to one of this finite number.Thus all forms with a given discriminant can be segregated into classes,each class consisting of forms equivalent to one member of that class.This result and some inductively cstablished results by Legendre attracted Gauss's attention. In a bold step Gauss extracted from Lagrange's work the notion of equiva- lence of forms and concentrated on that.The fifth section of his disouisitiones by far the largest section,is devoted to this subject 25.Noup Mim.e1'Acad.de Berlin,1773,263-312;and1775,323fE.=Eurs,3,693-795
THE THEORY OF FORMS 827 Gauss systematized and extended the theory of forms.He first defined equivalence of forms.Let F=ax2 2bxy cy? be transformed by means of(4)into the form F'=a'x'2+2b'x'y'+c'y'2 Then b'2-d=(b2-ac)(a6-y)2 If now (8-By)2=1,the discriminants of the two forms are cqual.Then the inverse of the transformation (4)will also contain integral coefficients (by Cramer's rule)and will transform F'into F.F and F are said to be cquivalent.If -By =1,F and F are said to be properly equivalent, and if a8-By =-1,then F and F'are said to be improperly equivalent. Gauss proved a number of theorems on the equivalence of forms.Thus if Fis equivalent to F and F'to F"then F is equivalent to F".If F is equiva- lent toF,any number M representable by Fis representable byand in as many ways by one as by the other.He then shows,if Fand Fare cquiva- lent.how to find all the transformations from finto f de also finds all the representations of a given number M by a form F,provided the values of and y are relatively prime. By definition two equivalent forms have the same value for their discriminant D However,two forms with the same discriminant are not necessarily equivalent.Gauss shows that all forms with a given D can be segregated into classes;the members of any one class are properly equivalent to each other.Though the number of forms with a given D is infinite,the number of classes for a given D is finite.In cach class one form can be taken as representative and Gauss gives criteria for the choice of a simplest representative.The simplest form of all those with determinant D has a =1,b=0,c=-D.This he calls the principal form and the class to which it belongs is called the principal class. Gauss then takes up the composition (product)of forms.If the form F=AX2+2BXY+CY2 is transformed into the product of two forms f=ax2 2bxy +cy? and f=a'x'2b'x'y'c'y'a by the substitution X=pxx+p2y+px'y十pg Y=91xx'+gaxy'gax'y +qayy