by E. T. Bell
Another of Kronecker’s teachers at the Gymnasium also influenced him profoundly and became his lifelong friend, Ernst Eduard Kummer (1810-1893), subsequently professor at the University of Berlin and one of the most original mathematicians Germany has produced, of whom more will be said in connection with Dedekind. These three, Kronecker senior, Werner, and Kummer, capitalized Leopold’s immense native abilities, formed his mind, and charted the future course of his life so cunningly that he could not have departed from it if he had wished.
Already in this early stage of his education we note an outstanding feature of Kronecker’s genial character, his ability to get along with people and his instinct for forming lasting friendships with men who had risen in the world or were to rise, and who would be useful to him either in business or mathematics. This genius for friendships of the right sort, which is one of the successful business man’s distinguishing traits, was one of Kronecker’s more valuable assets and he never mislaid it. He was not consciously mercenary, nor was he a snob; he was merely one of those lucky mortals who is more at ease with the successful than with the unsuccessful.
Kronecker’s performance at school was uniformly brilliant and many-sided. In addition to the Greek and Latin classics which he mastered with ease and for which he retained a lifelong liking, he shone in Hebrew, philosophy, and mathematics. His mathematical talent appeared early under the expert guidance of Kummer, from whom he received special instruction. Young Kronecker however did not concentrate to any great extent on mathematics, although it was obvious that his greatest talent lay in that field, but set himself to acquiring a broad liberal education commensurate with his manifold abilities. In addition to his formal studies he took music lessons and became an accomplished pianist and vocalist. Music, he declared when he was an old man, is the finest of all the fine arts, with the possible exception of mathematics, which he likened to poetry. These many interests he retained throughout his life. In none of them was he a mere dabbler: his love of the classics of antiquity bore tangible fruit in his affiliation with Graeca, a society dedicated to the translation and popularization of the Greek classics; his keen appreciation of art made him an acute critic of painting and sculpture, and his beautiful house in Berlin became a rendezvous for musicians, among them Felix Mendelssohn.
Entering the University of Berlin in the spring of 1841, Kronecker continued his broad education but began to concentrate on mathematics. Berlin at that time boasted Dirichlet (1805-1859), Jacobi (18041851) and Steiner (1796-1863) on its mathematical faculty; Eisenstein (1823-1852), the same age as Kronecker, also was about, and the two became friends.
The influence of Dirichlet on Kronecker’s mathematical tastes (particularly in the application of analysis to the theory of numbers) is clear all through his mature writings. Steiner seems to have made no impression on him; Kronecker had no feeling for geometry. Jacobi gave him a taste for elliptic functions which he was to cultivate with striking originality and brilliant success, chiefly in novel applications of magical beauty to the theory of numbers.
Kronecker’s university career was a repetition on a larger scale of his years at school: he attended lectures on the classics and the sciences and indulged his bent for philosophy by profounder studies than any he had as yet undertaken, particularly in the system of Hegel. The last is emphasized because some curious and competent reader may be moved to seek the origin of Kronecker’s mathematical heresies in the abstrusities of Hegel’s dialectic—a quest wholly beyond the powers of the present writer. Nevertheless there is a strange similarity between some of the weird unorthodoxies of recent doubts concerning the self-consistency of mathematics—doubts for which Kronecker’s “revolution” was partly responsible—and the subtleties of Hegel’s system. The ideal candidate for such an undertaking would be a Marxian communist with a sound training in Polish many-valued logic, though in what incense tree this rare bird is to be sought God only knows.
Following the usual custom of German students, Kronecker did not spend all his time at Berlin but moved about. Part of his course was pursued at the University of Bonn, where his old teacher and friend Kummer had taken the chair of mathematics. During Kronecker’s residence at Bonn the University authorities were in the midst of a futile war to suppress the student societies whose chief object was the fostering of drinking, duelling, and brawling in general. With his customary astuteness, Kronecker allied himself secretly with the students and thereby made many friends who were later to prove useful.
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Kronecker’s dissertation, accepted by Berlin for his Ph.D. in 1845, was inspired by Kummer’s work in the theory of numbers and dealt with the units in certain algebraic number fields. Although the problem is one of extreme difficulty when it comes to actually exhibiting the units, its nature can be understood from the following rough de scription of the general problem of units (for any algebraic number field, not merely for the special fields which interested Kummer and Kronecker). This sketch may also serve to make more intelligible some of the allusions in the present and subsequent chapters to the work of Kummer, Kronecker, and Dedekind in the higher arithmetic. The matter is quite simple but requires several preliminary definitions.
The common whole numbers 1, 2, 3, . . . are called the (positive) rational integers. If m is any rational integer, it is the root of an algebraic equation of the first degree, whose coefficients are rational integers, namely x—m = 0. This, among other properties of the rational integers, suggested the generalization of the concept of integers to the “numbers” defined as roots of algebraic equations. Thus if r is a root of the equation
xn + a1xn-1 + . . . + an-1x + an = 0,
where the a’s are rational integers (positive or negative), and if further r satisfies no equation of degree less than n, all of whose coefficients are rational integers and whose leading coefficient is 1 (as it is in the above equation, namely the coefficient of the highest power, xn, of x in the equation is 1), then r is called an algebraic integer of degree n. For example, is an algebraic integer of degree 2, because it is a root of x2 −2x + 6 = 0, and is not a root of any equation of degree less than 2 with coefficients of the prescribed kind; in fact is the root of and the last coefficient, is not a rational integer.
If in the above definition of an algebraic integer of degree n we suppress the requirement that the leading coefficient be 1, and say that it can be any rational integer (other than zero, which is considered an integer), a root of the equation is then called an algebraic number of degree n. Thus is an algebraic number of degree 2, but is not an algebraic integer; it is a root of 2x2—2x + 3 = 0.
Another concept, that of an algebraic number field of degree n is now introduced: if r is an algebraic number of degree n, the totality of all expressions that can be constructed from r by repeated additions, subtractions, multiplications, and divisions (division by zero is not defined and hence is not attempted or permitted), is called the algebraic number field generated by r, and may be denoted by F[r]. For example, from r we get r + r, or 2r; from this and r we get 2r/r or 2, 2r—r or r, 2r × r or 2r2, etc. The degree of this F[r] is w.
It can be proved that every member of F[r] is of the form c0rn-l +c1rn-2 + . . .+ cn-1, where the c’s are rational numbers, and further every member of F[r] is an algebraic number of degree not greater than n (in fact the degree is some divisor of n). Some, but not all, algebraic numbers in F[r] will be algebraic integers.
The central problem of the theory of algebraic numbers is to investigate the laws of arithmetical divisibility of algebraic integers in an algebraic number field of degree n. To make this problem definite it is necessary to lay down exactly what is meant by “arithmetical divisibility,” and for this we must understand the like for the rational integers.
We say that one rational integer, m, is divisible by another, d, if we can find a rational integer, q, such that m = q × d; d (also q) is called a divisor of m. For example 6 is a divisor of 12, because 12 = 2 × 6; 5 is not a divisor of 12 be
cause there does not exist a rational integer q such that 12 = q × 5.
A (positive) rational prime is a rational integer greater than 1 whose only positive divisors are 1 and the integer itself. When we try to extend this definition to algebraic integers we soon see that we have not found the root of the matter, and we must seek some property of rational primes which can be carried over to algebraic integers. This property is the following: if a rational prime p divides the product a × b of two rational integers, then (it can be proved that) p divides at least one of the factors a, b of the product.
Considering the unit, 1, of rational arithmetic, we notice that 1 has the peculiar property that it divides every rational integer; −1 also has the same property, and 1,-1 are the only rational integers having this property.
These and other clues suggest something simple that will work, and we lay down the following definitions as the basis for a theory of arithmetical divisibility for algebraic integers. We shall suppose that all the integers considered lie in an algebraic number field of degree n.
If r, s, t are algebraic integers such that r = s × t, each of s, t is called a divisor of r.
If j is an algebraic integer which divides every algebraic integer in the field, j is called a unit (in that field). A given field may contain an infinity of units, in distinction to the pair 1, −1 for the rational field, and this is one of the things that breeds difficulties.
The next introduces a radical and disturbing distinction between rational integers and algebraic integers of degree greater than 1.
An algebraic integer other than a unit whose only divisors are units and the integer itself, is called irreducible. An irreducible algebraic integer which has the property that if it divides the product of two algebraic integers, then it divides at least one of the factors, is called a prime algebraic integer. All primes are irreducibles, but not all irreducibles are primes in some algebraic number fields, for example in as will be seen in a moment. In the common arithmetic of 1, 2, 3 . . . the irreducibles and the primes are the same.
In the chapter on Fermat the fundamental theorem of (rational) arithmetic was mentioned: a rational integer is the product of (rational) primes in only one way. From this theorem springs all the intricate theory of divisibility for rational integers. Unfortunately the fundamental theorem does not hold in all algebraic number fields of degree greater than one, and the result is chaos.
To give an instance (it is the stock example usually exhibited in text books on the subject), in the field we have
each of is a prime in this field (as may be verified with some ingenuity), so that 6, in this field, is not uniquely decomposable into a product of primes.
It may be stated here that Kronecker overcame this difficulty by a beautiful method which is too detailed to be explained untechnically, and that Dedekind did likewise by a totally different method which is much easier to grasp, and which will be noted when we consider his life. Dedekind’s method is the one in widest use today, but this does not imply that Kronecker’s is less powerful, nor that it will not come into favor when more arithmeticians become familiar with it.
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In his dissertation of 1845 Kronecker attacked the theory of the units in certain special fields—those defined by the equations arising from the algebraic formulation of Gauss’ problem to divide the circumference of a circle into n equal parts or, what is the same, to construct a regular polygon of n sides.
We can now close up one part of the account opened by Fermat.
In struggling to prove Fermat’s “Last Theorem” that xn + yn = zn is impossible in rational integers x, y, z (none zero) if n is an integer greater than 2, arithmeticians took what looks like a natural step and resolved the left-hand side, xn + yn, into its n factors of the first degree (as is done in the usual second course of school algebra). This led to the exhaustive investigation of the algebraic number field mentioned above in connection with Gauss’ problem—after serious but readily understandable mistakes had been made.
The problem at first was studded with pitfalls, into which many a competent mathematician and at least one great one—Cauchy—tumbled headlong. Cauchy assumed as a matter of course that in the algebraic number field concerned the fundamental theorem of arithmetic must hold. After several exciting but premature communications to the French Academy of Sciences, he admitted his error. Being restlessly interested in a large number of other problems at the time, Cauchy turned aside and failed to make the great discovery which was well within the capabilities of his prolific genius and left the field to Kummer. The central difficulty was serious: here was a species of “integers”—those of the field concerned—which defied the fundamental theorem of arithmetic; how reduce them to law and order?
The solution of this problem by the invention of a totally new kind of “number” appropriate to the situation, which (in terms of these “numbers”) automatically restored the fundamental theorem of arithmetic, ranks with the creation of non-Euclidean geometry as one of the outstanding scientific achievements of the nineteenth century, and it is well up in the high mathematical achievements of all history. The creation of the new “numbers”—so-called “ideal numbers”—was the invention of Kummer in 1845. These new “numbers” were not constructed for all algebraic number fields but only for those fields arising from the division of the circle.
Kummer too had fallen afoul of the net which snared Cauchy, and for a time he believed that he had proved Fermat’s “Last Theorem.” Then Dirichlet, to whom the supposed proof was submitted for criticism, pointed out by means of an example that the fundamental theorem of arithmetic, contrary to Kummer’s tacit assumption, does not hold in the field concerned. This failure of Kummer’s was one of the most fortunate things that ever happened in mathematics. Like Abel’s initial mistake in the matter of the general quintic, Kummer’s turned him into the right track, and he invented his “ideal numbers.”
Kummer, Kronecker, and Dedekind in their invention of the modern theory of algebraic numbers, by enlarging the scope of arithmetic ad infinitum and bringing algebraic equations within the purview of number, did for the higher arithmetic and the theory of algebraic equations what Gauss, Lobatchewsky, Johann Bolyai, and Riemann did for geometry in emancipating it from slavery in Euclid’s too narrow economy. And just as the inventors of non-Euclidean geometry revealed vast and hitherto unsuspected horizons to geometry and physical science, so the creators of the theory of algebraic numbers uncovered an entirely new light, illuminating the whole of arithmetic and throwing the theories of equations, of systems of algebraic curves and surfaces, and the very nature of number itself, into sharp relief against a firm background of shiningly simple postulates.
The creation of “ideals”—Dedekind’s inspiration from Kummer’s vision of “ideal numbers”—renovated not only arithmetic but the whole of the algebra which springs from the theory of algebraic equations and systems of such equations, and it proved also a reliable clue to the inner significance of the “enumerative geometry”I of Plücker, Cayley and others, which absorbed so large a fraction of the energies of the geometers of the nineteenth century who busied themselves with the intersections of nets of curves and surfaces. And last, if Kronecker’s heresy against Weierstrassian analysis (noted later) is some day to become a stale orthodoxy, as all not utterly insane heresies sooner or later do, these renovations of our familiar integers, 1, 2, 3, . . . , on which all analysis strives to base itself, may ultimately indicate extensions of analysis, and the Pythagorean speculation may envisage generative properties of “number” that Pythagoras never dreamed of in all his wild philosophy.
Kronecker entered this beautifully difficult field of algebraic numbers in 1845 at the age of twenty two with his famous dissertation De Unitatibus Complexis (On Complex Units). The particular units he discussed were those in algebraic number fields arising from the Gaussian problem of the division of the circumference of a circle into n equal arcs. For this work he got his Ph.D.
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nbsp; The German universities used to have—and may still have—a laudable custom in connection with the taking of a Ph.D.: the successful candidate was in honor bound to fling a party—usually a prolonged beer bust with all the trimmings—for his examiners. At such festivities a mock examination consisting of ridiculous questions and more ridiculous answers was sometimes part of the fun. Kronecker invited practically the whole faculty, including the Dean, and the memory of that undignified feast in celebration of his degree was, he declared in later years, the happiest of his life.
In at least one respect Kronecker and his scientific enemy Weierstrass were much alike: they were both very great gentlemen, as even those who did not particularly care for either admitted. But in nearly everything else they were almost comically different. The climax of Kronecker’s career was his prolonged mathematical war against Weierstrass, in which quarter was neither given nor asked. One was a born algebraist, the other almost made a religion of analysis. Weierstrass was large and rambling, Kronecker a compact, diminutive man, not over five feet tall, but perfectly proportioned and sturdy. After his student days Weierstrass gave up his fencing; Kronecker was always an expert gymnast and swimmer and in later life a good mountaineer.
Eyewitnesses of the battles between this curiously mismatched pair tell how the big fellow, annoyed by the persistence of the little fellow, would stand shaking himself like a good-natured St. Bernard dog trying to rid himself of a determined fly, only to excite his persecutor to more ingenious attacks, till Weierstrass, giving up in despair, would amble off, Kronecker at his heels still talking maddeningly. But for all their scientific differences the two were good friends, and both were great mathematicians without a particle of the “great man” complex that too often inflates the shirts of the would-be mighty.
