Leopold Kronecker

How to use this course

Leopold Kronecker is remembered by many students for a symbol, \(\delta_{ij}\), and by historians for a quarrel with Georg Cantor. Both memories miss the centre of his work. Kronecker was one of the great number theorists of the nineteenth century. He saw that roots of unity, elliptic functions and quadratic forms are parts of one arithmetic theory, and his conjectures on that unity set the agenda for class field theory. He also held a firm philosophy: mathematics should be built from the integers by finite, explicit constructions.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Kronecker’s own papers are dense and often leave proofs incomplete. We state his results in modern language and say clearly where later mathematicians finished what he began.

Timeline

Liegnitz and Kummer, 1823–1841

Leopold Kronecker was born on 7 December 1823 in Liegnitz, a town in Prussian Silesia that is now Legnica in Poland. His father, Isidor Kronecker, ran a successful business; his mother, Johanna Prausnitzer, came from a wealthy family. The family was Jewish. A younger brother, Hugo Kronecker, became a well-known physiologist.

Leopold was taught at home by private tutors before entering the Liegnitz Gymnasium. There his mathematics teacher was Ernst Kummer, then a young schoolmaster who would soon become one of the leading number theorists in Europe. Kummer saw the boy’s ability and encouraged him to do research of his own. The two became lifelong friends. Fifty years later, at Kummer’s doctoral jubilee, Kronecker said that he owed to Kummer the most essential part of his intellectual life.

The Gymnasium gave him a classical education as well, and he kept a strong interest in Greek and Latin literature and in philosophy all his life. This breadth mattered: his later views on what mathematics is, and which of its objects are real, grew out of serious philosophical reading.

Berlin, Bonn, Breslau, 1841–1845

Kronecker entered the University of Berlin in 1841. He attended lectures in philosophy, classical philology, natural science and mathematics. His mathematics teachers included Peter Gustav Lejeune Dirichlet, the founder of analytic number theory, the geometer Jakob Steiner and the astronomer Johann Encke.

Like many German students he moved between universities. He spent the summer of 1843 at Bonn, mostly studying astronomy, and joined a student fraternity there. In the winter of 1843–44 he went to Breslau, where Kummer had just been appointed professor. Kummer was then developing his theory of ideal numbers, which restores unique factorization in rings of cyclotomic integers. Kronecker absorbed this work at its source.

He returned to Berlin in 1844 and wrote his dissertation under Dirichlet. De unitatibus complexis (On complex units) studied the units of the rings of integers generated by roots of unity, a problem at the junction of Dirichlet’s and Kummer’s research. He submitted it on 30 July 1845 and passed the oral examination on 14 August. Georg Frobenius later compared the thesis to chemistry done without the atomic hypothesis, because it worked with cyclotomic integers before the ideal numbers that would clarify them were fully available.

The business years, 1845–1861

With a doctorate in hand, Kronecker did not seek an academic post. He returned to Liegnitz to manage the banking business of an uncle and a large family farming estate. He was good at it and became wealthy. In 1848 he married his cousin Fanny Prausnitzer, the daughter of his mother’s brother. They had six children.

Mathematics continued as a private passion. He corresponded with Kummer and worked on the theory of equations. In 1853 he published a short paper on equations solvable by radicals, extending Évariste Galois. It contains the statement, then unproved in general, that every abelian equation over the rationals has roots in a cyclotomic field (Lesson 05). In 1858, at the same time as Charles Hermite and Francesco Brioschi, he related the general quintic equation to elliptic modular functions.

By 1855 he no longer needed to live on the estate. He moved to Berlin as a private scholar with independent means. Berlin was then becoming the leading mathematical centre in Germany. Kummer arrived as professor in 1855, and Karl Weierstrass in 1856. Kronecker had no university post, but his papers on algebra, elliptic functions and number theory established him quickly. On 23 January 1861 the Berlin Academy elected him a member. Membership carried the right to lecture at the university, and from 1862 he did so regularly.

Units and roots of unity

For the general reader. In the ordinary integers only \(1\) and \(-1\) have inverses. In larger number systems, such as the numbers \(a+b\sqrt2\), infinitely many numbers have inverses, and they form a group with interesting structure. Kronecker’s thesis studied these units in rings built from roots of unity. A few years later he proved a striking fact: an algebraic integer all of whose conjugates lie on the unit circle must itself be a root of unity.

The unit group

Let \(K\) be a number field of degree \(n=r_1+2r_2\), with \(r_1\) real and \(r_2\) pairs of complex embeddings, and ring of integers \(\mathcal O_K\). The units \(\mathcal O_K^\times\) are the elements of norm \(\pm1\).

For \(K=\mathbb Q(\zeta_p)\) with \(p\) an odd prime, \(r_1=0\) and \(r_2=(p-1)/2\), so the free rank is \((p-3)/2\). Natural units come from the roots of unity themselves. For \(1\le k\le p-1\),

is an algebraic integer, and so is its inverse (Exercise 4.2). These cyclotomic units generate a subgroup of finite index in \(\mathcal O_K^\times\). Kummer later showed that this index equals the class number of the real subfield \(\mathbb Q(\zeta_p+\zeta_p^{-1})\), a result in the line of inquiry Kronecker opened.

Kronecker’s theorem on roots of unity (1857)

Proof. Let \(\alpha_1,\dots,\alpha_n\) be the roots. For each \(k\ge1\), the polynomial \(f_k(x)=\prod_i(x-\alpha_i^k)\) has integer coefficients, because they are symmetric functions of the \(\alpha_i\) with integer coefficients. Its \(j\)-th coefficient is bounded by \(\binom nj\) in absolute value, since every \(|\alpha_i^k|\le1\). There are only finitely many integer polynomials of degree \(n\) with such bounded coefficients, so the set \(\{\alpha_i^k\}\) of all roots of all \(f_k\) is finite. Hence for each \(i\) there are \(k<l\) with \(\alpha_i^k=\alpha_i^l\). Since \(\alpha_i\ne0\), \(\alpha_i^{l-k}=1\). \(\square\)

The theorem is the starting point of the study of heights. Lehmer’s question, whether the Mahler measure of a non-cyclotomic integer polynomial is bounded away from \(1\), is a quantitative form of it and remains open.

Abelian equations and the Kronecker–Weber theorem

For the general reader. Some polynomial equations have roots that can be written using only roots of unity, the complex numbers \(e^{2\pi i k/n}\). Kronecker found that this happens exactly when the symmetry group of the equation is commutative. His theorem is the first complete description of a whole class of number fields, and the model for one of the deepest theories in mathematics.

Cyclotomic fields

The field \(\mathbb Q(\zeta_m)\), \(\zeta_m=e^{2\pi i/m}\), is Galois over \(\mathbb Q\) with group

This group is abelian, so every subfield of a cyclotomic field is an abelian extension of \(\mathbb Q\). Kronecker’s claim was the converse.

History. Kronecker stated the theorem in 1853. His argument was incomplete for extensions of degree a power of 2. Heinrich Weber published a proof in 1886 that contained gaps, found and repaired by Olaf Neumann in 1981. Hilbert gave the first complete proof in 1896, using his theory of ramification groups.

The quadratic case

Let \(p\) be an odd prime and \(\left(\frac{a}{p}\right)\) the Legendre symbol. The Gauss sum is \(g=\sum_{a=1}^{p-1}\left(\frac ap\right)\zeta_p^a\in\mathbb Q(\zeta_p)\).

Proof sketch. Expand \(g^2=\sum_{a,b}\left(\frac{ab}{p}\right)\zeta^{a+b}\). Substitute \(b=ac\) to get \(\sum_c\left(\frac cp\right)\sum_a\zeta^{a(1+c)}\). The inner sum is \(p-1\) if \(c\equiv-1\) and \(-1\) otherwise. Since \(\sum_c\left(\frac cp\right)=0\), the total is \(\left(\frac{-1}{p}\right)p\).

With \(\sqrt{-1}=\zeta_4\) and \(\sqrt2=\zeta_8+\zeta_8^{-1}\), every quadratic field \(\mathbb Q(\sqrt d)\) lies in \(\mathbb Q(\zeta_{4|d|})\). The general theorem reduces, by a careful study of ramification, to cyclic extensions of prime-power degree ramified at a single prime; there the Galois structure of \(\mathbb Q(\zeta_{p^k})\) leaves no room for any other field.

Class field theory in miniature

The theorem says the maximal abelian extension of \(\mathbb Q\) is \(\mathbb Q(\zeta_\infty)=\bigcup_m\mathbb Q(\zeta_m)\), generated by special values of the exponential function \(e^{2\pi i x}\) at rational \(x\). The Galois group is \(\widehat{\mathbb Z}^\times=\varprojlim(\mathbb Z/m)^\times\), and the Frobenius at a prime \(\ell\nmid m\) is \(\sigma_\ell\). Kronecker asked what replaces the exponential function over other base fields (Lesson 06). Hilbert made that question his twelfth problem in 1900.

Complex multiplication and the Jugendtraum

For the general reader. Roots of unity are values of the exponential function. Kronecker’s favourite dream, which he called the dearest dream of his youth, was that values of elliptic functions play the same role for imaginary quadratic fields such as \(\mathbb Q(\sqrt{-5})\). He was right. The dream turned into the theory of complex multiplication, and its extension to other fields is still unsolved.

Lattices with extra symmetry

For \(\tau\) in the upper half-plane, let \(\Lambda_\tau=\mathbb Z+\mathbb Z\tau\) and \(E_\tau=\mathbb C/\Lambda_\tau\). The \(j\)-invariant \(j(\tau)\) classifies \(E_\tau\) up to isomorphism; it has the expansion \(j(\tau)=q^{-1}+744+196884q+\cdots\) with \(q=e^{2\pi i\tau}\). Usually the only complex numbers \(\alpha\) with \(\alpha\Lambda_\tau\subseteq\Lambda_\tau\) are integers. When \(\tau\) is an imaginary quadratic irrationality, the ring of such \(\alpha\) is an order in \(K=\mathbb Q(\tau)\), and \(E_\tau\) is said to have complex multiplication.

The second part adds torsion points: adjoining to \(H\) the values of a Weber function at the points of finite order of \(E_\tau\) produces all abelian extensions of \(K\). This is the full content of Kronecker’s Jugendtraum. He described it in a letter to Dedekind in 1880 and proved parts of it. The complete statement was established only after Teiji Takagi created class field theory around 1920. Goro Shimura and Yutaka Taniyama later extended the theory to CM fields.

Counting classes with forms

Kronecker worked with binary quadratic forms \(ax^2+bxy+cy^2\) of discriminant \(D=b^2-4ac<0\). A positive definite form is reduced if \(|b|\le a\le c\), with \(b\ge0\) when \(|b|=a\) or \(a=c\). Each class contains exactly one reduced form, and for fundamental \(D\) the number of reduced forms is \(h_K\), \(K=\mathbb Q(\sqrt D)\). Each reduced form gives a root \(\tau=(-b+\sqrt D)/2a\) and a conjugate \(j(\tau)\).

The Kronecker limit formula

For the general reader. Some infinite sums diverge, yet their divergence has a precise shape, and what remains after removing it carries information. Kronecker computed the leftover part of a sum over a two-dimensional lattice and found a classical function of elliptic function theory. The formula connects analysis to the arithmetic of quadratic fields.

The real-analytic Eisenstein series

For \(\tau=x+iy\) with \(y>0\) and \(\operatorname{Re}s>1\) define

It is invariant under \(\tau\mapsto(a\tau+b)/(c\tau+d)\) for \(\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\mathrm{SL}_2(\mathbb Z)\), and it continues meromorphically to all \(s\) with a simple pole at \(s=1\). Let \(\eta(\tau)=q^{1/24}\prod_{n\ge1}(1-q^n)\) be the Dedekind eta function.

Proof idea

Separate the terms with \(m=0\), which give \(2\zeta(2s)y^s\). For \(m\ne0\), apply Poisson summation in \(n\). The zero Fourier mode gives a multiple of \(\zeta(2s-1)y^{1-s}\), which carries the pole. The other modes give an exponentially convergent series. At \(s=1\) it equals

The term \(2\zeta(2)y=\pi^2y/3\) supplies \(-2\pi\log|q|^{1/12}\), and the Laurent expansion of the \(\zeta(2s-1)\) term at \(s=1\) supplies the pole, \(\gamma\), \(\log 2\) and \(\log\sqrt y\). Together they give \(-2\pi\log|\eta(\tau)|^2\) plus the stated constants.

Arithmetic use

For an imaginary quadratic field \(K\) of discriminant \(D\), the Dedekind zeta function \(\zeta_K(s)\) is a finite sum of values \(E(\tau_{\mathfrak a},s)\), one for each ideal class, times \((\sqrt{|D|}/2)^{-s}\) and a factor counting units. Comparing residues gives the class number formula. The constant terms give the Chowla–Selberg formula, which expresses periods of CM elliptic curves through values of the Gamma function. Erich Hecke built on Kronecker’s formulas in his work on zeta functions of number fields.

Divisors, forms and the Grundzüge

For the general reader. Kummer had restored unique factorization in special number rings by inventing “ideal numbers”. Dedekind turned these into sets of numbers called ideals. Kronecker disliked infinite sets and found another way: he represented an ideal by a polynomial whose coefficients generate it, and he computed with these polynomials. His 1882 treatise built the arithmetic of algebraic numbers and of algebraic functions on this basis.

The Grundzüge

Grundzüge einer arithmetischen Theorie der algebraischen Grössen was written for Kummer’s fiftieth doctoral jubilee in 1881 and published in Crelle’s Journal in 1882. It treats fields generated over \(\mathbb Q\) or over a field of rational functions in one framework, the “domains of rationality”. It proves the primitive element theorem and that degrees multiply in towers, and it requires algorithms, for example a test for irreducibility, before a concept counts as defined. Jean Dieudonné saw in it the first dream of a single theory joining arithmetic and algebraic geometry.

Adjoining a root without assuming it exists

The proof is the Euclidean algorithm. If \(g\notin(f)\), then \(\gcd(f,g)=1\), so \(af+bg=1\) for some \(a,b\in k[x]\), and \(\bar b\) is the inverse of \(\bar g\). Nothing is assumed about the complex numbers. This is the construction every algebra course now uses, and it fits Kronecker’s demand that mathematical objects be built by finite operations.

Forms and their contents

For a polynomial (form) \(F=\sum a_\alpha u^\alpha\) in indeterminates \(u_1,u_2,\dots\) with coefficients in \(\mathcal O_K\), Kronecker considered the “divisor” generated by its coefficients, which corresponds to Dedekind’s ideal \(c(F)=(a_\alpha)\). The ideal \((a_1,\dots,a_r)\) is thus represented by the single linear form \(a_1u_1+\cdots+a_ru_r\).

The theorem lets one multiply ideals by multiplying polynomials. Dedekind proved an equivalent statement in 1892: if all coefficients of \(FG\) are algebraic integers, then so is every product \(a_ib_j\) of a coefficient of \(F\) with a coefficient of \(G\). The method of adjoining indeterminates survives today in the Nagata ring \(R(u)\) and in the theory of Kronecker function rings.

Abelian groups and approximation

For the general reader. Two of Kronecker’s results have become standard tools far from where he found them. The first says that every finite commutative group is built from cyclic groups. The second says that if you step around a circle, or around a doughnut, by amounts that share no rational relation, you eventually come arbitrarily close to every point.

Finite abelian groups (1870)

In an 1870 paper on the class numbers of Kummer’s ideal complex numbers, Kronecker considered a finite set of elements with a commutative, associative composition in which cancellation holds. He did not use the word group, but this is an axiomatic definition of a finite abelian group, and he proved its structure theorem. Frobenius and Ludwig Stickelberger restated the theorem in group language in 1878.

Proof sketch. Reduce to a \(p\)-group. Take an element \(a\) of maximal order \(p^e\). One shows that \(\langle a\rangle\) has a complement: every coset of \(\langle a\rangle\) contains an element whose order equals the order of the coset in \(A/\langle a\rangle\). Induction on \(|A|\) then splits \(A/\langle a\rangle\), and the lifts give \(A\cong\langle a\rangle\times B\). Uniqueness follows by counting the elements of order dividing \(p^k\) for each \(k\).

Kronecker’s approximation theorem (1884)

Proof for \(n=1\). Let \(\alpha\) be irrational. Among the \(N+1\) numbers \(\{k\alpha\}\), \(0\le k\le N\), two lie in the same interval of length \(1/N\) by the pigeonhole principle. Their difference gives \(j\) with \(0<\|j\alpha\|<1/N\), where \(\|\cdot\|\) is the distance to the nearest integer; it is non-zero since \(\alpha\) is irrational. The multiples of \(j\alpha\) then step around the circle in increments smaller than \(1/N\), so every point is within \(1/N\) of some \(\{k\alpha\}\). Hermann Weyl’s equidistribution theorem of 1916 later strengthened density to uniform distribution.

Arithmetization and finitism

For the general reader. Kronecker believed that only the whole numbers are given to us, and that every other mathematical object must be constructed from them in finitely many steps. He accepted fractions and algebraic numbers, defined by explicit equations, and rejected arbitrary real numbers, infinite sets and proofs that show existence without construction. These views put him in conflict with much of the mathematics of his day, and made him a forerunner of twentieth-century constructivism.

The programme

In a lecture of 1886, according to Heinrich Weber, Kronecker said that the dear God made the integers and all else is the work of man. In Über den Zahlbegriff (1887) he set out the programme. Arithmetic, algebra and analysis, though not geometry and mechanics, were to be reduced to the natural numbers. Statements about other numbers were to become statements about integers and about polynomials with integer coefficients, read through congruences.

For example, an identity among Gaussian integers becomes a congruence of integer polynomials modulo \(x^2+1\): the symbol \(i\) is replaced by the indeterminate \(x\), and the relation \(i^2=-1\) by the modulus. This is the ring \(\mathbb Z[x]/(x^2+1)\) of Lesson 08, used here as a philosophical tool. Kronecker treated algebraic numbers in general in the same way.

What he rejected

In 1886 he attacked the theories of irrational numbers of Dedekind, Cantor and Eduard Heine, all of which used infinite sets or sequences of rationals as completed objects. He praised Ferdinand Lindemann’s 1882 proof that \(\pi\) is transcendental but asked what use it was, since in his view irrational numbers of that kind did not exist. He rejected Weierstrass’s continuous nowhere-differentiable function and Cantor’s transfinite numbers. Definitions had to be decidable: to say a polynomial is irreducible one must have a procedure that decides it in finitely many steps.

The method was found earlier by the astronomer Friedrich von Schubert in 1793, and it is hopelessly slow in practice. Its point is that irreducibility over \(\mathbb Z\) is decidable. Modern algorithms (reduction mod \(p\), Hensel lifting, lattice reduction) answer the same question efficiently. Hensel lifting is named after Kronecker’s student Kurt Hensel.

Power in Berlin, 1861–1891

For two decades Kronecker lectured at the University of Berlin without holding a professorship. His courses on number theory, the theory of algebraic equations, determinants and integrals drew able students, though many found them hard. He took part in the seminar with Kummer and Weierstrass, and the three made Berlin the leading school of mathematics in Germany. He declined the chair at Göttingen in 1868, preferring his position at the Berlin Academy. That year the Paris Academy elected him a corresponding member; the St Petersburg Academy followed in 1872, the London Mathematical Society made him an honorary member in 1875, and in 1884 he became a Foreign Member of the Royal Society.

In 1880, after Carl Borchardt’s death, he became editor of the Journal für die reine und angewandte Mathematik, known as Crelle’s Journal. In 1883 Kummer retired and Kronecker succeeded him in the chair, becoming co-director of the seminar with Weierstrass. His doctoral students included Franz Mertens, Kurt Hensel, Adolf Kneser, Paul Stäckel and Jules Molk. Hensel later edited his collected works.

Quarrels

His philosophy gave his editorial and academic power a sharp edge. In 1870 he tried to persuade Eduard Heine to withdraw a paper on trigonometric series. In 1877 he delayed the publication in Crelle’s Journal of Cantor’s paper on the equivalence of spaces of different dimensions. Cantor came to see him as a personal enemy (see the Cantor course). Weierstrass, once a friend, felt that Kronecker was persuading the next generation that his analysis had no value; in 1888 he considered leaving Berlin, but stayed because his successor would otherwise fall under Kronecker’s influence. Kronecker broke with Hermann Amandus Schwarz in 1885 over a joke about his small stature.

The quarrels should not hide his lifelong friendship with Kummer, his care for his students and his wide culture in philosophy and the classics.

Death and legacy

In the summer of 1891 Fanny Kronecker was seriously injured in a climbing accident. She died on 23 August. Kronecker converted from Judaism to Christianity in his last year. He died in Berlin on 29 December 1891, from complications of bronchitis, a few months after his wife. He is buried in the Alter St.-Matthäus-Kirchhof in Berlin-Schöneberg, where his grave is an honorary grave of the city. Kurt Hensel edited his Werke in five volumes (1895–1930) and published some of his lectures.

Legacy

• Class field theory: the Kronecker–Weber theorem and the Jugendtraum defined its goal. Hilbert’s twelfth problem asks for the analogue over every number field. It remains open, with recent progress by Samit Dasgupta and Mahesh Kakde for totally real fields.
• Algebra: the construction \(k[x]/(f)\), the structure of finite abelian groups and the arithmetic of forms belong to every algebra course. His divisor theory was revived in the twentieth century, notably by Harold Edwards in a constructive setting.
• Analysis and number theory: the limit formulas, the approximation theorem, the Kronecker symbol and Kronecker’s lemma in probability are standard tools. The Kronecker delta and Kronecker product carry his name in linear algebra.
• Foundations: his insistence on finite constructions anticipated the intuitionism of Brouwer and the constructive mathematics of Errett Bishop. Hilbert opposed him on this point, while carrying his number theory forward.

Connections

People linked to Leopold Kronecker across this collection. A name without a link has no page here yet. See Leopold Kronecker in the network of all 337 lives →

Teachers and advisors

• Ernst Kummer
• Peter Gustav Lejeune Dirichlet 1845

Students

• Adolf Kneser
• Franz Mertens
• Jules Molk
• Kurt Hensel
• Mathias Lerch
• Paul Stäckel

Influenced by

• Ernst Kummer 1830s-1891
• Niels Henrik Abel
• Évariste Galois

Influenced

• Adolf Hurwitz 1877-1882
• David Hilbert
• Georg Frobenius
• Hermann Minkowski
• Josiah Willard Gibbs 1867-1868
• Teiji Takagi

Colleagues and collaborators

• Charles Hermite
• Georg Frobenius
• Karl Weierstrass 1856-1891
• Wilhelm Weber

Rivals

• Camille Jordan 1874
• Georg Cantor 1877-1891
• Gösta Mittag-Leffler
• Richard Dedekind

Family

• Hugo Kronecker

Also mentioned in this life

• L. E. J. Brouwer
• Leonhard Euler
• Joseph Fourier
• Carl Friedrich Gauss
• Erich Hecke
• Adrien-Marie Legendre
• Siméon Denis Poisson
• Atle Selberg
• Jakob Steiner
• Hermann Weyl

Fields

• number theory 45 lives
• algebra 29 lives
• elliptic functions 4 lives
• foundations of mathematics 10 lives

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Leopold Kronecker, MacTutor History of Mathematics.
2. Leopold Kronecker, Wikipedia; German Wikipedia.
3. H. M. Edwards, Kronecker, Leopold, Dictionary of Scientific Biography.
4. H. Weber, Leopold Kronecker, Jahresbericht der DMV 2 (1893).
5. G. Frobenius, Gedächtnisrede auf Leopold Kronecker, Abh. Preuss. Akad. Wiss. (1893).

Mathematics

1. L. Kronecker, Werke, ed. K. Hensel, 5 vols., Teubner, 1895–1930.
2. L. Kronecker, Grundzüge einer arithmetischen Theorie der algebraischen Grössen, J. reine angew. Math. 92 (1882).
3. L. Kronecker, Über den Zahlbegriff, J. reine angew. Math. 101 (1887).
4. E. Penchèvre, On Kronecker’s Grundzüge, arXiv:1801.04327.
5. Kronecker–Weber theorem; Hilbert’s twelfth problem; Kronecker limit formula; Kronecker’s theorem, Wikipedia.
6. H. M. Edwards, Divisor Theory, Birkhäuser, 1990.
7. D. A. Cox, Primes of the Form \(x^2+ny^2\), 2nd ed., Wiley, 2013 (complex multiplication and the Jugendtraum).
8. O. Neumann, Two proofs of the Kronecker–Weber theorem “according to Kronecker, and Weber”, J. reine angew. Math. 323 (1981).