Vl duµ, Kronecker's Jugendtraum and modular functions

Chapter 1

Kronecker

The ideas and results presented in this book have been heavily in�uenced by the per-sonality of Leopold Kronecker. Kronecker's works have endured a rather complicatedhistory, and for reasons of both a personal and a mathematical nature their proper as-sessment was delayed until the middle of the twentieth century.

In this chapter we attempt to shed light on Kronecker's personality and on his creativeactivity and to describe the impact of Kronecker's ideas and mathematical philosophy onmodern mathematics.

1.1. Biography

Kronecker's life was outwardly uneventful. He was born in 1823 into a well-to-doJewish family living in Liegnitz. Kronecker had a private tutor until he entered theLiegnitz Gymnasium. At the Gymnasium his mathematics teacher was Eduard Kummer,later a prominent mathematician, who soon recognized the outstanding ability of hisstudent and encouraged him to pursue independent research. Their acquaintance initiateda friendship which lasted until Kronecker's death in 1891.

In 1841 Kronecker matriculated to the University of Berlin, where he attended lec-tures in mathematics given by Dirichlet and Steiner. His interests were not exclusivelyrestricted to mathematics; like Gauss and Jacobi, he was interested in classical philologyand attended lectures on this subject. Kronecker also attended Schelling's philosophy lec-tures; later he was to make a thorough study of the works of Descartes, Spinoza, Leibniz,Kant and Hegel as well as Schopenhauer, whose ideas he rejected. Kronecker maintainedhis interest in philosophy throughout his life.

During 1843-44 Kronecker spent one semester at the University of Bonn and anotherat the University of Breslau (now Wroclaw, Poland), largely because at that time his olderfriend Kummer was lecturing at these universities. In the winter of 1844-45 Kroneckerreturned to Berlin to write his doctoral dissertation entitled �De unitatibus complexis�(�On Complex Units�), which he submitted to the Faculty of Philosophy on July 30, 1845.On September 10 of the same year Kronecker was awarded a doctorate. From 1845 to1855 Kronecker lived on an estate near Liegnitz managing family business and pursuingmathematics as a recreation. In 1848 he married his cousin Fanny Prausnitzer; they even-tually had six children. In 1855 Kronecker returned to Berlin. In the same year Kummeralso moved to Berlin to succeed Dirichlet (who had been invited to Göttingen after thedeath of Gauss) in the chair of mathematics at the University. In 1856 Weierstrass alsocame to Berlin and Borchardt became the editor of Crelle's Journal. These events markedthe formation of the mathematical school of Berlin.

After returning to Berlin, Kronecker started intensive research in mathematics, par-ticularly in arithmetic, algebra, the theory of elliptic functions, and their interrelations.In 1860 Kummer, seconded by Borchardt and Weierstrass, nominated Kronecker to theBerlin Academy, of which he later became a full member on January 23, 1861.

In 1862 Kronecker started teaching at the University as an associate professor. Hislectures did not attract great numbers of students; the lectures were rather unpolished

1

because Kronecker often attempted to convey his fresh ideas. Kronecker's position allowedhim considerable freedom of scienti�c activity. For this reason, when in 1868 he was o�eredthe chair at Göttingen (held successively by Gauss, Dirichlet, and Riemann) he refusedit. He was to become a full professor of the University of Berlin in 1883, only eight yearsbefore his death.

Kronecker was an active and in�uential member of the Berlin Academy. Many ofhis works were reported at meetings of the Academy and appeared in its publications.Kronecker was also a member of many foreign academies (in 1868 he became a foreignmember of the Paris Academy and in 1884 a foreign member of the Royal Society ofLondon). Kronecker's activity was signi�cantly in�uenced by his travels abroad, and hisBerlin home was always a center of hospitality for foreign scientists.

In the middle of the 1870s disagreement began to develop between Kronecker andWeierstrass concerning fundamental issues of the philosophy of mathematics. Kroneckerrejected the foundations of Weierstrass's function theory as well as Cantor's views ontrans�nite numbers.

As a result of this discord, and for other reasons related to personal and mathematicalissues, a large part of Kronecker's heritage was recognized neither by his contemporaries,nor by mathematicians working in the �rst half of our century. Many of his ideas havebeen appreciated only during the past 40 years (see �3 and Weil's work [Wei 6]).

1.2. Main works

Kronecker was a productive and versatile mathematician. The complete collectionof his works [Kr 1], published by his devoted pupil Hensel in 1895-1930, consists of sixlarge volumes numbered 1-5, (the third volume being divided into two parts). Kronecker'sworks deal with various problems of arithmetic, algebra, function theory, analysis, andparticularly with their interrelations. It is, therefore, rather di�cult to give an exhaustivereview of Kronecker's results. Here we attempt to concentrate on those aspects of Kro-necker's creative work that appear to be most signi�cant for the science of today, and thatmost vividly re�ect his personality. In addition to the results discussed in this section,Kronecker established a number of others, some of which are referred to in Sections 2.6and 2.10.

For the sake of clarity, we divide Kronecker's results into separate sections relatingto di�erent mathematical disciplines, although for Kronecker, whose genius was revealedmost vividly in his study of the interrelations between di�erent disciplines, such a divisionwould have been rather arti�cial.

A) Algebra

Kronecker's basic achievements in algebra relate to linear algebra, �eld theory, and grouptheory. In linear algebra he completed the formulation of basic theorems on solvingsystems of linear equations (real or complex coe�cients were assumed, but the proofswere quite general). In his lectures at the University of Berlin, Kronecker suggested anaxiomatic characterization of the determinant as an antisymmetric function of n vectorsin an n-dimensional space, which equals 1 for the unit matrix. He also introduced thenotion of the tensor product of linear spaces and the �Kronecker� (tensor) product ofmatrices. Kronecker established a number of other results in algebra concerning thereduction of bilinear forms to canonical form, the computation of the Galois group ofthe modular equation, and the analysis of the canonical forms of square matrices. Theseresults, however, will not be discussed here.

2

In �eld theory, Kronecker suggested an important construction of the splitting �eld ofa polynomial

F (x) = xn + A1xn�1 + � � �+ An;

that is, an extension L of the �eld K = Q(A1; : : : ; An), such that the polynomial F splitsinto linear factors over L. Here the �eld L is constructed as the residue class �eld moduloan irreducible polynomial f(x) over K. Kronecker thus solves the problem of constructingan irreducible polynomial f(x) such that

F (x) = (x� �1) � � � (x� �n) mod f(x):

If the initial polynomial has rational coe�cients, one obtains the construction of an alge-braic number �eld Q(�) with f(�) = 0. This approach relates to Kronecker's general ideaof making all of mathematics �arithmetical� and is in keeping with the �constructivism�of his thought (see Section 3). Kronecker contributed signi�cantly to the developmentof group theory. In his doctoral dissertation he established a speci�c case of the Dirich-let theorem on units (for cyclotomic �elds); this was the �rst example of decomposinga �nitely generated abelian group into a direct sum of a �nite group and a free abeliangroup. Moreover, in his paper Auseinandersetzung einiger Eigenschaften der Klassen-zahl idealer complexer Zahlen (Exposition of some properties of the class number of idealcomplex numbers), 1870 [Kr 1, Vol. 11, 273-285], Kronecker, starting from the problemof studying the structure of the class group of a number �eld, suggested a procedure fordecomposing a �nite abelian group into a direct sum of cyclic groups, a technique whichis essentially used in modern manuals.

B) Number theory

In arithmetic Kronecker recorded quite a number of major achievements. These relatebasically to the application of profound algebraic or analytical ideas to number theory,but their signi�cance can be partly evaluated within the framework of arithmetic.

Kronecker's dissertation De unitatibus complexis written in 1845, is devoted to thestudy of the structure of units in a cyclotomic �eld, that is, the �eld obtained fromthe �eld of rationale by adding a root of unity. In this work, Kronecker provided acomplete description of the group of units, thereby proving an important speci�c case ofthe Dirichlet theorem on units in algebraic number �elds and obtaining one of the �rst(if not the very �rst) examples of decomposing a �nitely generated abelian group into adirect sum of cyclic groups.

The Kronecker-Weber theorem derived by Kronecker in 1853 (and completely provedby Weber much later), is essential to number theory. This theorem asserts that anyabelian extension of the �eld of rationals Q is contained in a cyclotomic �eld, that is, ina �eld obtained from Q by adding a root of unity.

The above arithmetical results attest to Kronecker's interest in general number-theoretic relationships and structures. On the other hand, Kronecker made e�orts toestablish speci�c number-theoretic relations and to compute particular number-theoreticfunctions. He would treat certain relations using methods within the framework of ageneral theory and would afterwards return to studying the same problems from an �ele-mentary� speci�c point of view. This approach is common for many outstanding mathe-maticians such as Fermat, Euler, and Gauss. Here we present one of the most importantexamples of the results obtained in this way by Kronecker.

The paper Über bilineare Formen mit vier Variabeln (On Bilinear Forms in FourVariables) 1883, [Kr 1, Vol.IV, 425-496] is entirely devoted to the arithmetical deduction

3

of results concerning arithmetic functions, developed earlier using the theory of elliptic andtheta functions. Some of these results were established by Kronecker himself, and someby other authors (for example Jacobi). In this paper Kronecker gave a purely number-theoretic proof of the so-called �Klassenzahlrelationen� (class number relations), some ofwhich relations he had established earlier on the basis of the complex multiplication ofelliptic functions (1857), others by means of theta functions (1860, 1862, 1875, see belowthe section devoted to the application of elliptic functions to number theory). In thissame paper, Kronecker also suggested a purely arithmetical proof of Jacobi's results onthe number of representations of an integer as the sum of four squares. He proved theFermat conjecture that every integer can be represented as the sum of three triangularnumbers, which is equivalent to determining which numbers can be expressed as the sumof three squares. Moreover, Kronecker gave an expression for the function N3(m), yieldingthe number of representations of an integer m as the sum of three squares.

For many years Kronecker studied the quadratic reciprocity law, investigating its his-tory, commenting on various proofs, and suggesting his own versions.

C) Theory of elliptic functions. . .

D) Application of elliptic functions to arithmetic and algebra. . .

E) Algebraic geometry

It is perhaps astonishing that algebraic geometry stands out as a speci�c area of Kro-necker's work. Kronecker's thought, as far as we can judge, was not of a geometric nature.On the contrary, he always tended to deal with algebraic and number-theoretic relations.Nevertheless, from the modern point of view one should classify certain important workof Kronecker as algebraic geometry.

We primarily mean the following works of 1881-82: Über die Discriminante algebrais-cher Functionen der einer Variabeln (On the Discriminant of Algebraic Functions of OneVariable), 1881 [Kr 1, Vol.II, 193-236], and Grundzüge einer arithmetischen Theorie deralgebraischen Grössen (Foundations of an Arithmetical Theory of Algebraic Values), 1882[Kr 1, Vol.II, 239-387]. These papers are devoted to the general division theory for alge-braic number �elds and algebraic function �elds in one variable (see Section 2.10). Thesubject treated by Kronecker is, however, broader. In modern parlance, he studies ide-als of an integral domain constituting a �nite algebra over one of the polynomial ringsC[x1; : : : ; xn] or Z[x1; : : : ; xn]. His objective was to decompose the variety generated byan ideal into irreducible components. To this end Kronecker uses an elimination methodwhich leads to the construction of a set of equations in (n�1) indeterminates (x2; : : : ; xn),whose solutions are exactly the projections of the solutions (x1; : : : ; xn) of the simultane-ous equations Fi(x1; : : : ; xn) = 0, i = 1; 2; : : : ; r, where Fi; : : : ; Fr make up a system ofgenerators of the ideal in question. Kronecker de�nes neither the dimension of a varietynor the notion of an irreducible variety and his arguments, therefore, often remain un-clear. Only in the case of the principal ideal do his arguments appear irreproachable andthey yield the proof that the rings C[x1; : : : ; xn] and Z[x1; : : : ; xn] are factorial. In 1905Lasker, however, showed that the elimination method can be applied to decomposing anyalgebraic variety into irreducible components.

One can consider the ideas presented in these papers from a somewhat di�erent pointof view, relating algebraic geometry to number theory. However, this may be done only in

4

the context of modern mathematics and thus we postpone this discussion until the nextsection.

1.3. The impact of Kronecker's ideas

Kronecker's mathematical genius was as vivid as it was original. Unfortunately, how-ever, he lacked the ability of presenting his profound ideals clearly. Partly because of thisand partly for certain mathematical reasons, such as the need for formalization, axiom-atization, rigorous mathematical thought, and clarity, the in�uence of Kronecker's ideason mathematics as a whole was for a long time felt only indirectly (for instance, throughthe works of his pupil Hensel). His genius was not recognized until as late as the middleof the twentieth century.

Here we shall brie�y summarize Kronecker's views on the philosophy of mathematicsand the in�uence of these views on modern mathematics.

Kronecker's views are quintessentially expressed in two ideas. The �rst is the re-jection of actual in�nity as a mathematical reality. The second is also prohibitive innature and can best be expressed by his well-known dictum �God Himself made theintegers-everything else is the work of men�. This leads to the prohibition against usingmathematical notions and methods which cannot be reduced to the arithmetic of integers.Despite their prohibitive form these statements carry a powerful positive charge.

The prohibition against using the actual in�nity in mathematical calculations forcedKronecker to make use of constructions which can simultaneously involve only a �nitenumber of elements. Kronecker made e�orts to turn the constructions he examined intoreal algorithms capable of producing appropriate objects rather than leaving them to beabstract notions. As an example one can consider Kronecker 's approach to the de�nitionof divisibility in general algebraic number �elds. This characteristic of Kronecker's thoughtis here particularly pronounced when his approach is compared with that suggested byDedekind (see Section 2.10). The divisibility de�nition in Dedekind's theory of idealsconcerns testing in�nitely many conditions, whereas Kronecker's divisibility criterion isintrinsically �nite.

The desire to argue �algorithmically� in�uenced Kronecker's works, many of which arebased on the method of computation of some mathematical objects. For example, thetreatise Grundzüge einer arithmetischen Theorie der algebraischen Grössen, which hasalready been mentioned, not only provides an �abstract� divisibility theory for algebraicnumber �elds, but also suggests a method of constructing the, theory of divisors for agiven �eld. This also relates to the problem of decomposing a variety into irreducibleones, for which a method was proposed requiring a �nite number of steps. Examinationof Kronecker's paper of 1886 on the theory of elliptic functions (see Sections 4.2 and4.3 below) also shows that Kronecker was arriving at his statements through painstakingcalculations. The �algorithmic� character of Kronecker's thought can be compared to thatof such predecessors as Leibniz, Euler, and Jacobi.

On the other hand, striving for arithmetization of mathematics allowed Kroneckerto obtain remarkable results. Among them it is worth mentioning the construction ofthe splitting �eld of a polynomial. This achievement owes its origins in part to theconstructivism requirement mentioned above; it is also linked with the desire to make thearithmetic of algebraic number �elds independent of the theory of complex numbers, whichis possible under an �abstract� de�nition of an algebraic number �eld (regarded without itsembeddings into the complex �eld). One might �nd other examples of Kronecker's results

5

which were a�ected by his desire for arithmetization, but we would rather concentratehere on the whole research program outlined in his treatise Grundzüge .... In �2 we havealready described the impact that this work had on algebraic geometry. However, itsin�uence on mathematics as a whole goes much further. According to Weil's opinion,which was advanced in his address to the Congress of Mathematicians in 1950 [Wei 8] onthe interrelation between number theory and algebraic geometry, Kronecker in Grundzügetried to lay the groundwork for a new area of mathematics which would cover both numbertheory and algebraic geometry as particular cases. Let us brie�y explain this opinion.

One of Kronecker's main ideas was the concept of specialization. For example, in hisimpressive work of 1886 concerning the theory of elliptic functions (which we consider inChapter 4 and reproduce in Part II of this book), Kronecker laid the technical foundationsfor the theory of complex multiplication of elliptic functions. He was, however, studyingarbitrary elliptic functions, approaching those with complex multiplication (the case ofthe so-called �singular moduli�) by the use of specialization. The notion of specialization,together with the remark that only a �nite number of points and varieties, whose common�eld of de�nition is �nitely generated over a prime �eld, are simultaneously involved inany speci�c problem of algebraic geometry, leads to the notion of absolutely algebraic�elds (that is, of �elds that are algebraic over their prime sub�elds), and shows their rolein algebraic geometry. After all, any statement of algebraic geometry can be reformu-lated as a theorem on algebraic varieties over a prime �eld. Recent works, presentingzero-characteristic counterparts of the methods initially used to prove the Riemann con-jecture for varieties over �nite �elds, are evidence that we can expect much from such anapproach. The specialization ideas lead us to the notion of the Kronecker dimension of avariety. Actually, Kronecker dimension coincides with the dimension of the correspondingGrothendieck scheme. The Kronecker dimension of a curve over an algebraic number �eldthus equals 2. This understanding of the dimension also leads to profound results forwhich we shall provide a validation in Part III of this book.

Weil's convincing argument shows that the concept of the height of a point of analgebraic variety over a number �eld and its major properties also belong to the circle ofKronecker's ideas.

It is particularly interesting to follow the in�uence of such ideas on the later devel-opment of mathematics. Kronecker's views have been further extended in intuitionismand constructivism. Also, striving for �arithmetization� of mathematics, which has beenparticularly pronounced in the development of algebraic geometry, has led to great suc-cesses in this area as well. Kronecker's ideas were adopted by Weil and enabled him tocreate �arithmetic algebraic geometry�, the term used by Weil. The Grothendieck theoryof schemes has completed the task set forth by Kronecker, since algebraic geometry andnumber theory are now branches of the same discipline.

We hope that a comparison of the contents of Parts I and III of this book will enablethe reader to assess the above-mentioned in�uence of Kronecker's ideas on modern math-ematics. Kronecker's in�uence can be traced in two directions. One is related to the factthat the theory of modular functions itself is largely the creation of Kronecker. The otheris that the general methods and problems discussed in Part III of our book are essentiallythe development of Kronecker's ideas concerning the �arithmetization� of mathematics.

6

Macaulay [19]

The algebraic theory of modular systems

Introduction

The present state of our knowledge of the properties of Modular Systems is chie�ydue to the fundamental theorems and processes of L. Kronecker, M. Noether, D. Hilbert,and K. Lasker, and above all to J. König's profound exposition and numerous extensionsof Kronecker's theory (p. xiii). König's treatise might be regarded as in some measurecomplete if it were admitted that a problem is �nished with when its solution has beenreduced to a �nite number of feasible operations. If however the operations are toonumerous or too involved to be carried out in practice the solution is only a theoreticalone; and its importance then lies not in itself, but in the theorems with which it isassociated and to which it leads. Such a theoretical solution must be regarded as apreliminary and not the �nal stage in the consideration of the problem.

In the following presentment of the subject Section I is devoted to the Resultant, thecase of n equations being treated in a parallel manner to that of two equations; Section IIcontains an account of Kronecker's theory of the Resolvent, following mainly the lines ofKönig's exposition; Section III, on general properties, is closely allied to Lasker's memoirand Dedekind's theory of Ideals; and Section IV is an extension of Lasker's results foundedon the methods originated by Noether. The additions to the theory consist of one or twoisolated theorems (especially 50 53 and 79 and its consequences; and the introduction ofthe Inverse System in Section IV.

7

Julius König [15]

Einleitung in die allgemeine Theorie der algebraischen Gröÿen

(Introduction to the general theory of algebraic magnitudes)

Part of Introduction

The whole presentation starts with the de�nition of �holoid� and �orthoid� domains,which reproduce the domains of rational integers vs. rational numbers, thus may, as itseems, be replaced by practicable technical expressions as domain of integrity and domainof rationality (�eld). The mindful reader will soon notice that this is not the case; forthose de�nitions avoid the rigidity of the latter concepts and permit consequently a muchsimpler foundation of the theory, abolish the unpleasant opposition between arithmeticand geometry, and account for the circumstance that the �orthoid� (the rational) is tobe considered as a special case of the �holoid� (the whole), which is important for theeconomy of the presentation. According to this terminology, the theory divides into an�algebraic� and an �arithmetic� part.

From the methodological point of view, I further wish to emphasise that the funda-mental theorem of Kronecker (chap. III. � 5�7) could be chosen as starting point of thewhole theory, by virtue of a completely elementary proof.

To this theorem may then be joined �as most important fundament of the resultsobtained here� the setup of the so-called resolvent form, which must be considered asan arithmetic extension of the concept of resultant that holds for an arbitrary system offorms, and may in particular always be represented as homogeneous linear form of thegiven forms. In doing so, the requirement of homogeneity is disregarded in the use of theexpression �form�, following Kronecker's example.

The introduction of the resolvent form on one hand, Kronecker's fundamental idea ofassociating new indeterminates on the other hand, lead to a general �in the full sense ofthe word� theory of elimination, in which the multiplicity of the manifolds de�ned by anysystem of equations is not neglected anymore, as it is the case in the �Festschrift�. Thusarises a powerful research tool, that yields �rstly a purely algebraic tool of functionaldeterminants. In a longer digression, a de�nitive presentation of the so-called specialelimination theory, that is the general theory of resultants and discriminants is given�for the latter for the �rst time.

The treatment of the linear diophantic problems provide a �rm fundament for the (inthe narrower sense of the word) arithmetic parts of the theory. By such is meant thegeneral solution of a system of equations whose single equations have the form

PFiXi =

F . In doing so, the F are considered as given and the X as unknown forms, which aresubject to the further condition that its coe�cients belong to a certain holoid domain thatis given beforehand. This problem is completely solved by a �nite, well de�ned sequenceof elementary operations in the cases that su�ce for the theory of algebraic magnitudes.These are the cases in which the coe�cients of the forms belong either to an orthoiddomain (thus for example any domain of rationality), or though to the domain of rationalintegers.

The �rst case yields among others a general treatment of Noether's theorem in thespace of n dimensions.

With these results the important but so far barely touched question about the equiv-alence of two divisor systems is not only completely solved, but also the more generalquestion of �containment� of a divisor system in another is settled.

8

In the theory of whole algebraic magnitudes, the two cases of (�absolutely�) wholemagnitudes in the narrow sense of general arithmetic and of (�relatively�) whole mag-nitudes with respect to an orthoid domain are treated simultaneously and by the samemethods. The second case contains among others the whole magnitudes within the mean-ing of function theory or geometry. It is a cardinal point of the theory that the idealmagnitudes are introduced from the outset as magnitudes capable not only of multiplica-tion but also of addition. An essentially new and simple method, resting above all on thetheory of the �module of equivalence�, builds on this fundament for the e�ective determi-nation of the fundamental system in all cases. The decomposition of a whole magnitudeinto prime ideals is carried out at last de�nitively and without case of exception, whereatthe corresponding results by Kronecker must be recti�ed in an essential point, becausethese are correct only in the simplest cases as a consequence of a noteworthy, indeed morefundamental mistake.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

First chapter.

Introductory fundamental concepts.

���

Number, magnitude, domain.

� 1. The domain of rational numbers, i.e., the embodiment of the positive ornegative, entire or fractional numbers, with inclusion of the zero, forms the fundament ofall research in the realm of pure mathematics.

The totality of the laws, according to which the operation on these numbers occursin the four species,�and only these with their corollaries�forms the embodiment of theproperties that we ascribe to the domain of rational numbers.

Number in the most general meaning of the word is called every concept whosecontent may be described completely by a series of rational numbers�or also in lastanalysis by a series of positive integers; the number is called�in the most general meaningof the word�algebraic, if this complete description can happen with the help of a finitenumber of integers.

This de�nition of the algebraic number and further below of the algebraic magnitudecan, according to the nature of the subject, only be provisional and lacunary; for it willindeed be the duty of the whole present book to construct the concept of the algebraicmagnitude and to develop its content. In any case, it is much too wide; for although thatcomplete description with the help of a �nite number of integers has in any case to beunderstood in the way that only their operations in the four species are to be understoodas properties of the latter, it is not at all ascertained of which kind those properties of thenew concept are, which enter into its complete description. Certain operations, analogousto the four species, will have to be declared, which however already demands for somedetailed explanations.

We designate a number as (mathematical) magnitude when some parts of its deter-mination, which naturally are themselves numbers, remain indeterminate. In the mostsimple case, an indeterminate entire or rational number is itself a magnitude. (Magni-tudes are further�if we anticipate the prospected conception�e.g., all entire functionsof the indeterminates x1; : : : ; xn with determinate number coe�cients). For the sake ofshorter wording, the concept of �magnitude� shall comprise also the �number�.

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An algebraic magnitude is thus materially determined by a �nite series of deter-minate and indeterminate integers. *)

The most general object of arithmetic-algebraic research can then be designated tobe the theory of those entire functions, whose coe�cients are themselves again algebraicmagnitudes.

The domain in which every single one of our investigations ranges, i.e., the totality ofall magnitudes that we overview at the same time, shall be a such one that in it certaincharacteristic properties of the domain of rational numbers remain preserved. Expressedmore precisely: Only such operations shal l be declared for the magnitudes ofthe considered domain whose formal laws coincide completely with those ofthe four species in the domain of rational numbers.

Then it is most convenient to designate those operations, insofar as they are at allpresent, also in the new domain as addition, subtraction, multiplication or division.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

*) The denominations magnitude and algebraic magnitude shall be used�following Kronecker�with this meaning. Outside the realm of arithmetic and algebra these expressions are used in a widermeaning. For instance every property of the things that may be described completely by numbers iscalled a magnitude. In function theory, one designates as algebraic functions also such magnitudes whichare de�ned by a �nite number of determinate and indeterminate numbers, without the need that they bealgebraic.

10

Viewpoint

Kronecker’sAlgorithmicMathematicsHAROLD M. EDWARDS

The Viewpoint column offers

mathematicians the opportunity to

write about any issue of interest to

the international mathematical

community. Disagreement and

controversy are welcome. The views

and opinions expressed here,

however, are exclusively those of the

author, and neither the publisher nor

the editor-in-chief endorses or accepts

responsibility for them. Viewpoint

should be submitted to the editor-in-

chief, Chandler Davis.

This essay is a lecture presented at‘‘Computability in Europe 2008,’’ Ath-ens, June 19, 2008.

IIwonder if it is as widely believed

by the younger generation ofmathematicians, as it is believed

by my generation, that Leopold Kro-necker was the wicked persecutor ofGeorg Cantor in the late nineteenthcentury and that, to the benefit ofmathematics, by the end of the centurythe views of Cantor had prevailed andthe narrow prejudices of Kroneckerhad been soundly and permanentlyrepudiated.

I suspect this myth persists whereverthe history of mathematics is studied,but even if it does not, an accurateunderstanding of Kronecker’s ideasabout the foundations of mathematics isindispensable to understanding con-structive mathematics, and the contrastbetween his conception of mathematicsand Cantor’s is at the heart of the matter.

It is true that he opposed the rise ofset theory, which was occurring in theyears of his maturity, roughly from 1870until his death in 1891. Set theory grewout of the work of many of Kronecker’scontemporaries—not just Cantor, butalso Dedekind, Weierstrass, Heine,Meray, and many others. However, asKronecker told Cantor in a friendly let-ter written in 1884, when it came to thephilosophy of mathematics he hadalways recognized the unreliability ofphilosophical speculations and hadtaken, as he said, ‘‘refuge in the safehaven of actual mathematics.’’ He wenton to say that he had taken great care inhis mathematical work ‘‘to express itsphenomena and truths in a form thatwas as free as possible from philo-sophical concepts.’’ Further on in thesame letter, he restates this goal of hiswork and its relation to philosophicalspeculations saying, ‘‘I recognize a truescientific value—in the field of mathe-matics—only in concrete mathematical

truths, or, to put it more pointedly, onlyin mathematical formulas.’’

Certainly, this conception of thenature and substance of mathematicsrestricts it to what is called ‘‘algorithmicmathematics’’ today, and it is what I hadin mind when I chose my title ‘‘Kro-necker’s Algorithmic Mathematics.’’Indeed, these quotations from Kro-necker show that my title is aredundancy—for Kronecker, thatwhich was not algorithmic was notmathematics, or, at any rate, it wasmathematics tinged with philosophicalconcepts that he wished to avoid.

At the time, I don’t think that thisattitude was in the least unorthodox.The great mathematicians of the firsthalf of the nineteenth century had, Ibelieve, similar views, but they hadfew occasions to express them,because such views were an under-stood part of the common culture.There is the famous quote from a letterof Gauss in which he firmly declaresthat infinity is a facon de parler andthat completed infinites are excludedfrom mathematics. According toDedekind, Dirichlet repeatedly saidthat even the most recondite theoremsof algebra and analysis could be for-mulated as statements about naturalnumbers. One needs only to open thecollected works of Abel to see that forhim mathematics was expressed, asKronecker said, in mathematical for-mulas. The fundamental idea of Galoistheory, in my opinion, is the theoremof the primitive element, whichallowed Galois to deal concretely withcomputations that involve the roots ofa given polynomial. And Kronecker’smentor Kummer—whom Kroneckercredits in his letter to Cantor withshaping his view of the philosophy ofmathematics—developed his famoustheory of ideal complex numbers in analtogether algorithmic way.

It is an oddity of history that Kro-necker enunciated his algorithms at atime when there was no possibility of

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Harold M. Edwards [11]

Kronecker's algorithmic mathematics.

Math. Intelligencer, 31(2):11�14, 2009.

11

implementing them in any nontrivialcases. The explanation is that thealgorithms were of theoretical, notpractical, importance to him. He goesso far as to say in his major treatiseGrundzuge einer arithmetischenTheorie der algebraischen Grossenthat, by his lights, the notion of irre-ducibility of polynomials lacks a firmfoundation (entbehrt einer sicherenGrundlage) unless a method is giventhat either factors a given polynomialor proves that no factorization ispossible.

When I first encountered this opin-ion of Kronecker’s, I had to read itseveral times to be sure I was not mis-understanding him. The opinion wasso different from my mid-twentiethcentury indoctrination in mathematicsthat I could scarcely believe he meantwhat he said. Imagine Bourbaki sayingthat the notion of an nonmeasurableset lacked a firm foundation until amethod was given for measuring agiven set or proving that it could notbe measured!

But he did mean what he saidand, as I have since learned, thereare other indications that the under-standing of mathematical thought inthat time was very different fromours. Another example of this isprovided by Abel’s statement in hisunfinished treatise on the algebraicsolution of equations that ‘‘at bottom’’(dans le fond) the problem of findingall solvable equations was the sameas the problem of determining whe-ther a given equation was solvable. Itwould be explicable if he had said

that the proof that an equation issolvable is ‘‘at bottom’’ the problemof solving it, but he goes much fur-ther: If you know how to decidewhether any given equation is solv-able, you know how to find allequations that are solvable.

To be honest, I don’t feel I fullyunderstand these extremely construc-tive views of mathematics—I am aproduct of my education—but knowingthat a mathematician of Abel’s caliberand experience saw mathematics in thisway is an important phenomenon that aviable philosophy of mathematicsneeds to take into account.

So Kronecker did mean it when hesaid that a method of factoring poly-nomials with integer coefficients isessential if one is to make use of irre-ducible polynomials, and he took careto outline such a method. I won’t gointo any explanation of his method—Idoubt that it was original with him, buthis treatise is the standard reference—except to say that it is pretty impracti-cal even with modern computers andto say that in his day it was utterly outof the question even for quite smallexamples.

This observation makes it indisput-able that the objective of Kronecker’salgorithm had to do with the meaningof irreducibility, not with practicalfactorization. It is a distinction that atfirst seems paradoxical but that arisesin many contexts. If you are trying tofind a specific root of a specific poly-nomial, Newton’s method is almostcertainly the best approach, but if youwant to prove that every polynomial

has a complex root, Newton’s methodis useless. In practice, it converges veryrapidly, but the error estimates are sounwieldy that you can’t prove that itwill converge at all until you are ableto prove that there is a root for it toconverge to, and for this you need amore plodding and less effectivemethod.

More generally, we all know that inpractical calculations clever guessworkand shortcuts can play important roles,and Monte Carlo methods are every-where. These are important topics inalgorithmic mathematics, but not inKronecker’s algorithmic mathematics. Iam not aware of any part of his workwhere he shows an interest in practicalcalculation. Again, his interest was inmathematical meaning, which for himwas algorithmic meaning.

I have always fantasized that Eulerwould be ecstatic to have access tomodern computers and would have awonderful time figuring out what hecould do with them, factoring Fermatnumbers and computing Bernoullinumbers. Kronecker, on the otherhand, I think would be much coolertoward them. In my fantasy, he wouldfeel that he had conceived of the cal-culations that interested him and had noneed to carry them out in any specificcase. His attitude might be the oneGalois expressed in the ‘‘preliminarydiscourse’’ to his treatise on the alge-braic solution of equations: ‘‘... I needonly to indicate to you the methodneeded to answer your question, with-out wanting to make myself or anyoneelse carry it out. In a word, the

....................................................................................................

AU

TH

OR HAROLD M. EDWARDS was a founding co-editor of The Mathe-

matical Intelligencer in 1978. Now Emeritus Professor at New York

University, he has lived in New York since graduating from the

University of Wisconsin in 1956, except for five years at Harvard and

one year at the Australian National University. He has received botha Steele Prize and a Whiteman Prize from the AMS. He lives in

Manhattan with his wife, journalist and author Betty Rollin.

Courant Institute of Mathematical Sciences

New York University

New York

NY 10012

USA

e-mail: edwards@cims.nyu.edu

12 THE MATHEMATICAL INTELLIGENCER

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calculations are impractical.’’ (... jen’aurai rien a y faire que de vous in-diquer le moyen de repondre a votrequestion, sans vouloir charger ni moi nipersonne de le faire. En un mot lescalculs sont impraticables.) This some-what provocative statement was omit-ted from the early publications ofGalois’s works. See page 39 of thecritical edition (1962) of Galois’s math-ematics, like Kronecker’s, wasalgorithmic but not practical. That’swhy it is not so surprising that all of thisalgorithmic mathematics—we couldcall it impractical algorithmic mathe-matics—was developed at a time whencomputers didn’t exist.

This, in my opinion, was Kroneck-er’s conception of mathematics—thatwhich his predecessors had accom-plished and that which he wanted toadvance. What generated the oncom-ing tide of set theory that was about toengulf this conception?

Kronecker wrote about the risingtendency in very few places, but whenhe did write about it, he identified themotive for its development: Set theorywas developed in an attempt toencompass the notion of the mostgeneral real number.

In 1904, after Kronecker had beendead for more than a dozen years,Ferdinand Lindemann published areminiscence about Kronecker that hasbecome a part of the Kronecker legendand that is surely wrong. According toLindemann, Kronecker asked him,apparently in a jocular way, ‘‘What isthe use of your beautiful researchesabout the number p? Why think aboutsuch problems when irrational num-bers do not exist?’’

We can only guess what Kroneckersaid to Lindemann that Lindemannremembered in this way, but I amconfident that he would not have saidthat irrational numbers did not exist.To be persuaded of this, one onlyneeds to know that Kronecker refers inhis lectures on number theory (theones edited and published by KurtHensel) to ‘‘the transcendental numberp from geometry,’’ which he describesby the formula p

4 ¼ 1� 13þ 1

5� 17þ � � � :

Note that Kronecker introduces p in hisfirst lecture on number theory. Notealso that he accepts p not only as an

irrational number but as a transcen-dental number; the proof of thetranscendence of p was of course theachievement for which Lindemannwas, and remains, famous. (His laterbelief that he had proved Fermat’s LastTheorem is benignly neglected.)

Kronecker, as one of the greatmasters of analytic number theory,made frequent use of transcendentalmethods and would have had noqualm about real numbers. Hisqualm—and he stated it explicitly—had to do with the conception of themost general real number.

My colleague Norbert Schappacherof the University of Strasbourg has dis-covered a document that statesKronecker’s qualm about the mostgeneral real number in a different wayand confirms Kronecker’s statement toCantor that his notions about the phi-losophy of mathematics were taughthim by Kummer. The document is aletter of Kummer in which he states thathe and Kronecker are in agreement intheir belief that the effort to createenough individual points to fill out acontinuum—that is, enough real num-bers to fill out a line—is as vain as theancient efforts to prove Euclid’s parallelpostulate. (The quotation occurs in aletter from Kummer to his son-in-law H.A. Schwarz, dated March 15, 1872, in theNachlass Schwarz of the archives of theBerlin-Brandenburg Academy of Sci-ences, folder 977.)

In our time, when young studentsare routinely told that ‘‘the real line’’consists of uncountably many realnumbers and that it is ‘‘complete’’ as atopological set, this opinion of Kum-mer and Kronecker is heresy in themost literal sense—it denies the truthof what young people are told has theagreement of all authorities.

So Kronecker, along with Kummer,saw what was going on—saw the pushto describe the most general realnumber, saw, as it were, the wish onthe part of his colleagues to talk about‘‘the set of all real numbers.’’ Moreover,he responded to it. His response was:It is unnecessary.

I have said that Kronecker says verylittle about the foundations of mathe-matics in his writings. But in the fewwords he does say, this message is

clear: It is unnecessary. One of themain goals of his mathematical workwas to demonstrate that it was unnec-essary by, as he told Cantor, expressingthe truths and phenomena of mathe-matics in ways that were as free aspossible from philosophical concepts.That would most certainly exclude anygeneral theory of real numbers. Hewished to show such a theory wasunnecessary by doing without it.

In view of the Kummer passagefound by Schappacher, we see that healso believed there was a specialimportance to his belief that the con-struction of the set of all real numberswas not necessary, because hebelieved it was doomed to fail.

In all likelihood you are now hear-ing for the first time the opinion that‘‘the real line’’ may not be a well-foun-ded concept, so I probably have norealistic hope of convincing you thatthis view may be justified. I won’t makea serious effort to do so. I will let it passwith just a brief reference to complica-tions like Russell’s paradox, Godel’sincompleteness theorem, the indepen-dence of the continuum hypothesis andthe axiom of choice, nonstandardmodels of the real numbers, and, com-ing at it from a different direction,Brouwer’s free choice sequences.There is a long history of unsuccessfulefforts to wrestle with infinity in a rig-orous way, efforts which, so far as Ihave ever been able to see, have beenconsistently frustrated. As Kummer andKronecker foresaw.

But even if one accepts that one dayit will succeed—or that it long ago didsucceed, except for uninteresting nit-picking—it seems to me that Kro-necker’s main message is still worthhearing and considering: It is unnec-essary. Mathematics should proceedwithout it to the maximum extentpossible. Kronecker was confident thatin the end its exclusion would prove tobe no impediment at all.

Well, of course modern mathemat-ics has painted itself into a corner inwhich dealing with infinity in a rigor-ous manner is necessary. If mathematicsis defined to be that which mathema-ticians do, then dealing with the realline is essential to mathematics. Ifmathematics insists on talking about

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‘‘properties of the real line’’ as thoughthe real line were a given, there is noroom for the belief that it isunnecessary.

Inevitably, then, Kronecker’s asser-tion is an assertion about the nature anddomain of mathematics itself. It assertsthat that which lies outside the Kro-neckerian conception of mathematics isunnecessary. (Instead of the Kro-neckerian conception, I would prefer tocall it the classical conception of math-ematics in deference to Euler and Gaussand Dirichlet and Abel and Galois, butsomehow ‘‘classical mathematics’’ hascome to mean the Cantorian opposite ofthis; therefore I am forced to call it theKroneckerian conception.)

With this meaning of ‘‘Kronecker’salgorithmic mathematics’’ in mind, wecan perhaps agree that it is unneces-sary to attempt to embrace the mostgeneral real number—to embrace ‘‘thereal line.’’ What is lost by adopting thisview of mathematics?

I often hear mention of what mustbe ‘‘thrown out’’ if one insists thatmathematics needs to be algorithmic.What if one is throwing out error?Wouldn’t that be a good thing ratherthan the bad thing the verb ‘‘to throwout’’ insinuates? I personally am notprepared to argue that what is beingthrown out is error, but I think one canmake a very good case that a gooddeal of confusion and lack of clarityare being thrown out.

The new ways of dealing withinfinity that set theory brought intomathematics can be seen in themethod used to construct an integralbasis in algebraic number theory.Kronecker gave an algorithm for thisconstruction. You could write a com-puter program following his plan, andthe program would work, although itmight be very slow. Hilbert in hisZahlbericht approaches the sameproblem in a different, and outra-geously nonconstructive, way. Heimagines all numbers in the field writ-ten as polynomials with rationalcoefficients in a particular generatingelement a. The polynomials are then ofdegree less than m, where m is the

degree of a. Moreover, there is acommon denominator for all the inte-gers in the field when they are writtenin this way. Hilbert has the chutzpah tosay: For each s = 1, 2, …, m, choose aninteger in the field which is repre-sented as a polynomial of degree lessthan s, and in which the numerator ofthe leading coefficient is the greatestcommon divisor of all numerators thatoccur in such integers. Such a choice isto be carried out for each s; the mintegers in the field ‘‘found’’ in this wayare an integral basis.

Let me try to state in as simple a wayas possible the process he is indicating:The integers in the field are a count-able set, so it is legitimate to regardthem as listed in an infinite sequence.The entries in the sequence are polyno-mials in a of degree less than m whosecoefficients are rational numbers witha fixed denominator D. For each s,Hilbert wants us to first strike from thelist all polynomials of degree s orgreater, and, from among those thatremain, choose one in which thenumerator of the coefficient of as-1 isnonzero, but otherwise is as small aspossible in absolute value. (Hilbertlooks at the greatest common divisorof the numerators rather than theabsolute value, but the effect is thesame.) So, not once but m times, weare to survey an infinite list of integersand pick out a nonzero one that hasthe smallest possible absolute value.

To put this in perspective, let medescribe an analogous situation.Imagine an infinite sequence of zerosand ones is given by some unknownrule. Would it be reasonable for me toask you to record a 1 if the sequencecontains infinitely many ones andotherwise to record a 0? In twentieth-century mathematics, one was asked todo such things all the time. Therefore,it is perhaps difficult to deny, as Iwould like to do, that it is a reasonablething to ask. But surely no one wouldcontend that it is an algorithm.

No doubt Hilbert regarded hisas a simplification of Kronecker’s con-struction. But only someoneindoctrinated in the nonconstructive

Hilbertian orthodoxy, as I was, and asmany of you surely were, could hear itcalled a ‘‘construction’’ without leapingfrom his or her chair in protest.

To ‘‘throw out’’ from mathematicsarguments of this type should be regar-ded as ridding it of ideas that are at bestsloppy thinking and at worst delusions.And in this particular case, the argumentfor throwing out Hilbert’s argument is allthe stronger because Kronecker hadalready shown many years earlier that itwas, in truth, unnecessary.

This contrast, between Kronecker’salgorithm for constructing an integralbasis and Hilbert’s nonconstructiveproof (can it be called a proof?) of theexistence of an integral basis, illustratesthe fork in the road that mathematicsencountered at the end of the nine-teenth century: To follow Kronecker’salgorithmic path, or to choose insteadthe daring new set-theoretic path pro-posedbyDedekind,Cantor,Weierstrass,and Hilbert.

You all understand very well whichpath was taken and you all understandas well how I feel about the choice thatwas made.

But now, in the twenty-first century, Ihope mathematicians will begin toreconsider that fateful choice. Now thatthere are conferences devoted to‘‘Computability in Europe’’ and mathe-maticians in their daily practice aredealingmore andmore with algorithms,approaching problems more and moreby asking themselves how they can usetheir powerful computers to gain insightand find solutions, the climate of opin-ion surely will change. How can anyonewho is experienced in serious compu-tation consider it important to conceiveof the set of all real numbers as amathematical ‘‘object’’ that can in someway be ‘‘constructed’’ using pure logic?For computers, there are no irrationalnumbers, so what reason is there toworry about the most general realnumber? Let us agree with Kroneckerthat it is best to express our mathematicsin a way that is as free as possible fromphilosophical concepts.Wemight in theend find ourselves agreeing with himabout set theory. It is unnecessary.

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Hermann Weyl [11]

Algebraic Theory of Numbers.

Princeton University Press

1940

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16

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Saunders MacLane [18]

History of abstract algebra:

origin, rise, and decline of a movement

The First Phase: Rings and Ideals

Emmy Noether

a short extract

Abstract algebra, as a conscious discipline, starts with Emmy Noether's 1921 paper�Ideal Theory in Ring Domains.� This paper sets out to examine systematically the de-composition of ideals in commutative rings with a chain condition. Her materials were athand: the term �ring� and number theoretic examples from Hilbert and some beginningabstract investigation of rings in a 1915 paper by Fraenkel. More important, questionsof invariant theory and algebraic geometry had led Lasker (1905) and Macaulev (1913,1916) to study, in a computational style, the decomposition of ideals in polynomial rings.Fraulein Noether's paper operated from the start more conceptually, not in polynomialrings but in arbitrary commutative rings with an ascending chain condition. At the begin-ning she observes that the familiar decomposition n = q1q2 � � � qr. of a rational integer ninto powers qi of distinct primes can be described conceptually in four di�erent ways: Asa representation of n in which (1) no two qi have a common factor; (2) the qi are relativelyprime in pairs; (3) the qi are primary; (4) the qi are irreducible. The paper goes on toestablish the four corresponding types of decomposition for ideals, in particular, the nowfamiliar representation of an ideal as an intersection of primary ideals (corresponding tothe geometric representation of an algebraic manifold as a union of irreducible manifolds).

At the time (1921), Emmy Noether had been publishing mathematical papers forabout eight years. Her doctoral dissertation (Erlangen, 1907) had been written underProfessor Gordan, an expert on invariant theory with a very computational emphasis. In1916, she moved to Göttingen as an assistant to Hilbert. The famous 1921 paper, justcited, marks dearly the point where she adopted whole heartedly a conceptual view ofalgebra. This was not done in isolation; the 1921 paper came shortly after a joint paper(1920) with W. Schmeidler on modules in non-commutative domains. It was followedsoon after by her write up of the thesis of Hentzelt (1923) on the theory of polynomialideals and resultants. This thesis had been submitted at Erlangen by Hentzelt in 1914;Hentzelt himself died shortly afterward in the war. Apparently, the original thesis washighly computational, but Noether's reformulation of the thesis emphasized her intent �togive this paper in purely conceptual formulation so that the ideas will become clearer.�There we have a clear description of the program of abstract algebra.

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Clémence Durvye and Grégoire Lecerf [5]

A concise proof of the Kronecker

polynomial system solver from scratch

From the introduction

Therein, the new central ingredients were the Kronecker representation of the varieties(originally due to Kronecker in [44], see de�nition in Section 3) and the idea of the liftedcurves . . . .

The new algorithm was programed in the Magma computer algebra system, and wascalled Kronecker [46] in homage to Leopold Kronecker for his seminal work about theelimination theory. The complete removal of the intermediate straight-line programs ledto the following features: only the input system needs to be represented by a straight-lineprogram, and the algorithm handles polynomials in at most two variables over the ground�eld.

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References

[1] B. Buchberger. Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischenGleichungssystems. Aequationes Math., 4:374�383, 1970.

[2] B. Buchberger. A theoretical basis for the reduction of polynomials to canonicalforms. ACM SIGSAM Bull., 10(3):19�29, 1976.

[3] David Cox, John Little, and Donal O'Shea. Ideals, varieties, and algorithms. Un-dergraduate Texts in Mathematics. Springer, New York, third edition, 2007. Anintroduction to computational algebraic geometry and commutative algebra.

[4] David E. Dobbs and Marco Fontana. Kronecker function rings and abstract Riemannsurfaces. J. Algebra, 99(1):263�274, 1986.

[5] Clémence Durvye and Grégoire Lecerf. A concise proof of the Kronecker polynomialsystem solver from scratch. Expo. Math., 26(2):101�139, 2008.

[6] Harold M. Edwards. Kronecker's views on the foundations of mathematics. In Thehistory of modern mathematics, Vol. I (Poughkeepsie, NY, 1989), pages 67�77. Aca-demic Press, Boston, MA, 1989.

[7] Harold M. Edwards. Kronecker's arithmetical theory of algebraic quantities. Jahres-ber. Deutsch. Math.-Verein., 94(3):130�139, 1992.

[8] Harold M. Edwards. Kronecker on the foundations of mathematics. In From Dedekindto Gödel (Boston, MA, 1992), volume 251 of Synthese Lib., pages 45�52. KluwerAcad. Publ., Dordrecht, 1995.

[9] Harold M. Edwards. Kummer and Kronecker. InMathematics in Berlin, pages 61�69.Birkhäuser, Berlin, 1998.

[10] Harold M. Edwards. Kronecker's fundamental theorem of general arithmetic. InEpisodes in the history of modern algebra (1800�1950), volume 32 of Hist. Math.,pages 107�116. Amer. Math. Soc., Providence, RI, 2007.

[11] Harold M. Edwards. Kronecker's algorithmic mathematics. Math. Intelligencer,31(2):11�14, 2009.

[12] David Eisenbud. Commutative algebra, volume 150 of Graduate Texts in Mathemat-ics. Springer-Verlag, New York, 1995. With a view toward algebraic geometry.

[13] Marco Fontana and K. Alan Loper. An historical overview of Kronecker functionrings, Nagata rings, and related star and semistar operations. In Multiplicative idealtheory in commutative algebra, pages 169�187. Springer, New York, 2006.

[14] Franz Halter-Koch. Lorenzen monoids: a multiplicative approach to Kronecker func-tion rings. Comm. Algebra, 43(1):3�22, 2015.

[15] Julius König. Einleitung in die allgemeine Theorie der algebraischen Gröÿen. B. G.Teubner, Leipzig, 1903.

[16] Leopold Kronecker. Grundzüge einer arithmetischen Theorie der algebraischenGrössen. (Abdruck einer Festschrift zu Herrn E. E. Kummers Doctor-Jubiläum, 10.September 1881.). J. reine angew. Math., 92(1):1�122, 1882.

[17] Leopold Kronecker. Leopold Kronecker's Werke. Bände I�V. Herausgegeben auf Ve-ranlassung der Königlich Preussischen Akademie der Wissenschaften von K. Hensel.Chelsea Publishing Co., New York, 1968.

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[18] Saunders Mac Lane. History of abstract algebra: origin, rise, and decline of a move-ment. In American mathematical heritage: algebra and applied mathematics (El Paso,Tex., 1975/Arlington, Tex., 1976), volume 13 of Math. Ser., pages 3�35. Texas TechUniv., Lubbock, Tex., 1981.

[19] F. S. Macaulay. The algebraic theory of modular systems. Cambridge MathematicalLibrary. Cambridge University Press, Cambridge, 1994. Revised reprint of the 1916original, With an introduction by Paul Roberts.

[20] Ray Mines, Fred Richman, and Wim Ruitenburg. A course in constructive algebra.Universitext. Springer-Verlag, New York, 1988.

[21] Fred Richman. Constructive aspects of Noetherian rings. Proc. Amer. Math. Soc.,44:436�441, 1974.

[22] Abraham Seidenberg. What is Noetherian? Rend. Sem. Mat. Fis. Milano, 44:55�61(1975), 1974.

[23] Mark van Atten, Göran Sundholm, Michel Bourdeau, and Vanessa van Atten. �Queles principes de la logique ne sont pas �ables�: Nouvelle traduction française annotéeet commentée de l'article de 1908 de L. E. J. Brouwer. Revue d'Histoire des Sciences,67(2):257�281, 2014.

[24] S. G. Vl duµ. Kronecker's Jugendtraum and modular functions, volume 2 of Studiesin the Development of Modern Mathematics. Gordon and Breach Science Publishers,New York, 1991. Translated from the Russian by M. Tsfasman.

[25] Hermann Weyl. Algebraic theory of numbers. Princeton Landmarks in Mathemat-ics. Princeton University Press, Princeton, NJ, 1998. Reprint of the 1940 original,Princeton Paperbacks.

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