Ernst Eduard Kummer

How to use this course

Ernst Eduard Kummer taught in a provincial grammar school for ten years before anyone offered him a university post. He went on to discover that unique factorization fails in rings of cyclotomic integers, to repair it with “ideal numbers”, and to prove Fermat’s Last Theorem for a large class of prime exponents. The concept of an ideal, the class number and much of modern algebraic number theory start here. He also left lasting work on hypergeometric functions and on algebraic surfaces, and he trained more doctoral students than almost any mathematician of his century.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Kummer wrote in the language of “complex numbers” built from roots of unity. The lessons translate his results into the modern language of rings and ideals that Richard Dedekind created to explain them.

Timeline

Sorau and Halle, 1810–1831

Ernst Eduard Kummer was born on 29 January 1810 in Sorau, a town in the Prussian province of Brandenburg that is now Żary in Poland. His father, Carl Gotthelf Kummer, was a physician who died in 1813, when Eduard was three. His mother brought up Eduard and his elder brother in hardship. Eduard was tutored privately and entered the Sorau gymnasium at nine.

In 1828 he went to the University of Halle to study Protestant theology. Mathematics was part of the course, and it was meant to serve philosophy. The lectures of Heinrich Ferdinand Scherk changed his plans. Scherk worked on algebra and series, and Kummer made mathematics his main subject.

In 1831 Scherk set a prize question for students, and Kummer’s essay on the expansion of powers of sines and cosines in terms of multiple angles won it. The essay, De cosinuum et sinuum potestatibus secundum cosinus et sinus arcuum multiplicium evolvendis, also served as his dissertation, and Halle awarded him a doctorate on 10 September 1831. In the same year he passed the state examination for gymnasium teachers.

The Liegnitz schoolmaster, 1832–1842

After a probationary year at his old school in Sorau, Kummer was appointed in 1832 to the gymnasium in Liegnitz, in Silesia, where he taught mathematics and physics for ten years. He was by every account an outstanding teacher, clear and warm, and he gave able pupils problems of real research interest. Two of them became mathematicians. Ferdinand Joachimsthal later held chairs in Halle and Breslau. Leopold Kronecker became Kummer’s lifelong friend and correspondent and, from 1855, his colleague in Berlin.

Kummer’s own research in these years was in analysis. His 1836 paper on the hypergeometric series (Lesson 04) appeared in Crelle’s Journal. He sent a copy to Carl Gustav Jacob Jacobi in Königsberg, which began a correspondence with Jacobi and with Peter Gustav Lejeune Dirichlet in Berlin. Both recognized him at once. On Dirichlet’s proposal the Berlin Academy elected the schoolteacher a corresponding member in 1839.

In 1840 he married Ottilie Mendelssohn, daughter of Nathan Mendelssohn and a cousin of Dirichlet’s wife Rebecka. The marriage tied him to the Berlin circle that would shape his career.

Breslau, 1842–1855

With support from Jacobi and Dirichlet, Kummer was appointed full professor at the University of Breslau in 1842. Here his interests moved to number theory. He studied Jacobi’s lectures on cyclotomy and the higher reciprocity laws, and he set out to understand arithmetic in the rings generated by roots of unity.

A letter to Kronecker of April 1844 shows him believing that every prime \(q\equiv1\pmod\lambda\) splits into \(\lambda-1\) prime factors among the cyclotomic integers. Jacobi, back from Italy that summer, showed him this was false. Kummer’s Latin memoir of 1844, written for the three-hundredth anniversary of the University of Königsberg, recorded the failure of unique factorization. Over the next year he found the cure. By October 1845 he had explained his theory of ideal prime factors to Kronecker, and the full papers appeared in 1846 and 1847 (Lessons 05 and 06). His own stated motivation was cyclotomy and the reciprocity laws; Fermat’s Last Theorem became central only in 1847 (Lesson 07).

Ottilie died in 1848, leaving young children. Kummer soon married Bertha Cauer, a cousin of Ottilie’s. He had thirteen children in all. In 1845 Breslau gave the young Gotthold Eisenstein an honorary doctorate; the Mathematics Genealogy Project lists Kummer as his advisor.

The hypergeometric series

For the general reader. Gauss had studied a single power series containing logarithms, inverse trigonometric functions and many special functions of physics. Kummer, still a schoolteacher, found how this series transforms: one function can be written in two dozen ways, each convenient in a different region. He also introduced the limiting “confluent” case now called Kummer’s function.

Gauss’s series

With \((a)_n=a(a+1)\cdots(a+n-1)\), define for \(|z|\lt1\) and \(c\notin\{0,-1,-2,\dots\}\)

which solves \(z(1-z)w''+[c-(a+b+1)z]w'-ab\,w=0\). Examples: \(\log(1+z)=z\,{}_2F_1(1,1;2;-z)\) and \((1-z)^{-a}={}_2F_1(a,b;b;z)\).

The six arguments are the images of \(z\) under the Möbius transformations permuting \(\{0,1,\infty\}\), a group isomorphic to \(S_3\). Each argument comes with four choices of exponents at the singular points, giving \(6\times4=24\). Riemann later derived all of this from his P-symbol.

Proof sketch of Pfaff. Euler’s integral \({}_2F_1(a,b;c;z)=\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}\int_0^1t^{b-1}(1-t)^{c-b-1}(1-zt)^{-a}dt\), valid for \(\operatorname{Re}c\gt\operatorname{Re}b\gt0\), becomes the right-hand side after the substitution \(t\mapsto1-t\). Analytic continuation removes the restriction on \(b,c\).

The confluent case

Replacing \(z\) by \(z/b\) and letting \(b\to\infty\) merges two singular points and gives Kummer’s function

with Kummer’s transformation \(M(a,c,z)=e^z\,M(c-a,c,-z)\). Bessel functions, Laguerre and Hermite polynomials, and the hydrogen-atom wave functions are special cases.

Cyclotomic integers

For the general reader. The ordinary integers factor uniquely into primes. Gauss showed the same is true for the integers \(a+bi\) with \(i^2=-1\). Kummer studied numbers built from a root of unity \(\zeta\) with \(\zeta^p=1\), because they split the expression \(x^p+y^p\) into linear factors. He discovered that for some \(p\) these numbers do not factor uniquely.

The ring

Let \(p\) be an odd prime and \(\zeta=e^{2\pi i/p}\). The minimal polynomial of \(\zeta\) is \(\Phi_p(x)=x^{p-1}+\cdots+x+1\), so \(\mathbb Q(\zeta)\) has degree \(p-1\) with Galois group \((\mathbb Z/p)^\times\), acting by \(\sigma_a:\zeta\mapsto\zeta^a\). Its ring of integers is \(\mathbb Z[\zeta]\). Kummer called its elements complex numbers \(f(\alpha)\). The norm

is a non-negative integer, multiplicative, and equals 1 exactly for units. Since \(\Phi_p(x)=\prod_a(x-\zeta^a)\), we have \(N(x-\zeta)=\Phi_p(x)\) for every integer \(x\).

The prime \(p\)

Put \(\pi=1-\zeta\). Then \(\Phi_p(1)=p=\prod_a(1-\zeta^a)\), and each \((1-\zeta^a)/(1-\zeta)=1+\zeta+\cdots+\zeta^{a-1}\) is a unit. Hence \(p=u\,\pi^{p-1}\) with \(u\) a unit: \(p\) is totally ramified. The quotient \(\mathbb Z[\zeta]/(\pi)\) is the field \(\mathbb F_p\).

Lamé’s factorization

Fermat’s equation factors as

If \(\mathbb Z[\zeta]\) had unique factorization, and the factors were pairwise coprime, each would be a unit times a \(p\)-th power, and a descent would follow. This was Gabriel Lamé’s plan in 1847. But unique factorization holds in \(\mathbb Z[\zeta_p]\) only for \(p\le19\) (a theorem of Montgomery and Uchida, 1971). For \(p=23\) the class number is 3: the prime 2, whose order mod 23 is 11, splits into two prime ideals of norm \(2^{11}\), neither of which is principal.

Ideal numbers

For the general reader. When a number system lacks unique factorization, Kummer’s remedy was to invent new “ideal” prime factors that do not exist as numbers, but whose divisibility can be tested. With them, factorization is unique again. Dedekind later turned each ideal prime into a concrete set, an ideal, and the idea became one of the foundations of modern algebra.

Decomposition

In particular \(q\) splits completely iff \(q\equiv1\pmod p\), and stays prime iff \(q\) is a primitive root mod \(p\). In Dedekind’s language, if \(\Phi_p\equiv\prod_{i=1}^g h_i\pmod q\), the prime ideals over \(q\) are \(\mathfrak q_i=(q,h_i(\zeta))\).

Ideal primes as tests

Kummer defined an ideal prime by the test it applies. For \(q\equiv1\pmod p\), pick a root \(r\) of \(\Phi_p\) modulo \(q\). Say the ideal prime \(\mathfrak q_r\) divides \(\alpha=f(\zeta)\) when \(f(r)\equiv0\pmod q\). Higher powers are detected by a refined test. The exponent of \(\mathfrak q_r\) in \(\alpha\) is a valuation \(v_{\mathfrak q_r}(\alpha)\), and Kummer proved that an element is determined up to a unit by its valuations at all ideal primes. Dedekind replaced \(\mathfrak q_r\) by the set of all \(\alpha\) it divides, the ideal \((q,\zeta-r)\).

Kummer proved the finiteness, introduced the class number of cyclotomic fields, and found an analytic formula for it by Dirichlet’s methods. He wrote \(h_p=h_p^+h_p^-\), where \(h_p^+\) is the class number of the real subfield, and computed the “first factor” \(h_p^-\) for all primes up to about 100.

Fermat’s Last Theorem for regular primes

For the general reader. On 1 March 1847 Gabriel Lamé announced a proof of Fermat’s Last Theorem to the Paris Academy. Joseph Liouville immediately asked whether the unique factorization it assumed was justified. Within weeks Kummer, in Breslau, had both the answer and a remedy: his ideal numbers prove Fermat’s claim for every “regular” prime, and he gave a simple test for regularity using Bernoulli numbers.

The episode

Liouville spoke after Lamé at the same meeting. In April 1847 Kummer sent his new proof for certain exponents to Dirichlet and wrote to Liouville. On 24 May 1847 Liouville read Kummer’s letter to the Academy. It enclosed the 1844 memoir showing that unique factorization fails, and announced the ideal numbers that repair it. Lamé’s approach was finished.

Proof of the first case

Suppose \(x^p+y^p=z^p\) with \(x,y,z\) coprime and \(p\nmid xyz\), and \(p\ge5\). Then the ideals \((x+\zeta^jy)\) are pairwise coprime: a common prime ideal would divide \((1-\zeta)y\) and \((1-\zeta)x\), hence \((1-\zeta)\), hence \(z\), forcing \(p\mid z\). Their product is \((z)^p\), so by unique factorization of ideals \((x+\zeta y)=\mathfrak a^p\) for some ideal \(\mathfrak a\). The class of \(\mathfrak a\) has order dividing \(p\) and dividing \(h_p\). Regularity forces \(\mathfrak a\) to be principal: \(x+\zeta y=u\,\alpha^p\) with \(u\) a unit.

Two facts finish the argument. Every \(\alpha^p\) is congruent to a rational integer modulo \(p\). Every unit is a root of unity times a real unit (Kummer’s lemma on units). Comparing \(x+\zeta y\) with its complex conjugate modulo \(p\) gives a linear relation among \(1,\zeta,\zeta^{-1},\zeta^{2}\) that is impossible unless \(p\mid x\) or \(p\mid y\). The second case, \(p\mid xyz\), needs a deeper lemma: for regular \(p\), a unit congruent to a rational integer modulo \(p\) is a \(p\)-th power.

Irregular primes

The irregular primes below 100 are 37, 59 and 67: \(37\mid B_{32}\), \(59\mid B_{44}\), \(67\mid B_{58}\). In 1857 Kummer extended his methods to these three exponents under additional hypotheses; later authors found and repaired gaps. K. L. Jensen proved in 1915 that there are infinitely many irregular primes. Whether there are infinitely many regular primes is still unknown, though about 61% of primes are expected to be regular. The full theorem waited for Andrew Wiles in 1995.

Bernoulli congruences and Kummer theory

For the general reader. Kummer noticed that Bernoulli numbers, divided by their index, repeat modulo a prime in a regular pattern. A century later this observation became the foundation of p-adic analysis of the zeta function. His study of root extensions, made for the reciprocity laws, became Kummer theory, a cornerstone of class field theory.

Since \(\zeta(1-k)=-B_k/k\) for even \(k\ge2\), the congruences say that \(k\mapsto(1-p^{k-1})\zeta(1-k)\) is \(p\)-adically continuous. Tomio Kubota and Heinrich-Wolfgang Leopoldt used this in 1964 to construct the \(p\)-adic zeta function, and Kenkichi Iwasawa built his theory of cyclotomic fields on it. The link between irregular primes and \(p\)-adic \(L\)-functions is the theme of the Herbrand–Ribet theorem.

Kummer theory

Key step. For \(\sigma\in\operatorname{Gal}(K(\alpha)/K)\) with \(\alpha^n=a\), \(\sigma(\alpha)/\alpha\) is an \(n\)-th root of unity, and the map \(\sigma\mapsto\sigma(\alpha)/\alpha\) is an injective homomorphism into \(\mu_n\) because \(\mu_n\subseteq K\) is fixed. The converse, that every cyclic extension of degree \(n\) is of this form, follows from Hilbert’s Theorem 90.

The Kummer surface

For the general reader. In his Berlin years Kummer turned to geometry: systems of light rays and the surfaces they envelop. In 1864 he described a surface of degree four with sixteen singular points, the largest number possible. Its sixteen points and sixteen special planes form a perfectly balanced configuration. The surface turned out to be a shadow of a four-dimensional torus, and it is the first example of what are now called K3 surfaces.

The torus picture

Let \(A=\mathbb C^2/\Lambda\) be a two-dimensional complex torus that is an abelian surface, such as the Jacobian of a genus-2 curve. The involution \(\iota(z)=-z\) has fixed points exactly at the 2-torsion points \(A[2]\cong(\mathbb Z/2)^4\), sixteen in all. The quotient \(A/\iota\) is a surface with 16 nodes, and the theta functions of second order embed it in \(\mathbb P^3\) as a quartic: a Kummer surface. Translations by \(A[2]\) commute with \(\iota\) and act on the configuration, which explains its symmetry. Fresnel’s wave surface in optics is a special case.

K3 surfaces

Blowing up the 16 nodes replaces each by a smooth rational curve and gives a smooth surface \(X\) with trivial canonical bundle and no holomorphic 1-forms. André Weil named such surfaces K3 in 1958, after Kummer, Kähler and Kodaira, and the mountain K2.

Series and binomial coefficients

For the general reader. Two smaller results of Kummer appear in every course. One is a flexible test for whether an infinite series of positive terms adds up to a finite number. The other tells you, by simple arithmetic in base \(p\), exactly how many times a prime divides a binomial coefficient.

Proof of convergence. If \(K_n\ge c\gt0\) for \(n\ge N\), then \(c\,a_{n+1}\le p_na_n-p_{n+1}a_{n+1}\). Summing telescopes: \(c\sum_{n=N}^{M}a_{n+1}\le p_Na_N\), so the partial sums are bounded. For divergence, \(K_n\le0\) means \(p_na_n\) is eventually non-decreasing, so \(a_n\ge C/p_n\). Taking \(p_n=1\) gives the ratio test; \(p_n=n\) gives Raabe’s test; \(p_n=n\log n\) gives Bertrand’s test.

Proof. Legendre’s formula gives \(v_p(N!)=\frac{N-s_p(N)}{p-1}\), where \(s_p\) is the base-\(p\) digit sum. Hence \(v_p\binom{m+n}{m}=\frac{s_p(m)+s_p(n)-s_p(m+n)}{p-1}\). Each carry reduces the digit sum by \(p\) in one place and adds 1 in the next, a net loss of \(p-1\). So the numerator is \(p-1\) times the number of carries. The hero figure shows the case \(p=2\); Lucas’s theorem (1878) is the special case of no carries.

Berlin, 1855–1883

In 1855 Dirichlet left Berlin for Göttingen to succeed Gauss. Kummer was offered the Berlin chair and accepted. Karl Weierstrass joined the university in 1856. Kronecker, wealthy and independent, also settled in Berlin in 1855, and later lectured at the university by right of his Academy membership. The three made Berlin the leading centre of mathematics in the world for two decades. In 1861 Kummer and Weierstrass founded the first seminar in Germany devoted to pure mathematics.

Kummer was an admired lecturer, clear and lively, and his courses drew large audiences. He was also a conscientious administrator: dean in 1857–58 and 1865–66, rector of the university in 1868–69, and from 1863 to 1878 permanent secretary of the Academy’s physical-mathematical class. From 1855 to 1874 he also taught at the War College, and his interest in ballistics led to papers on air resistance to projectiles in the 1870s.

In 1857 the Paris Academy awarded him its Grand Prix des Sciences Mathématiques for his work on Fermat’s theorem, although he had not entered the competition. It made him a correspondent in 1860 and a foreign associate in 1868. The Royal Society elected him a foreign member in 1863. In 1859 he joined Borchardt and Weierstrass in proposing Bernhard Riemann for the Berlin Academy.

Students

Kummer supervised or co-supervised dozens of doctorates; the Mathematics Genealogy Project lists 58. Among them were Georg Cantor, Hermann Amandus Schwarz (who married his daughter Marie), Lazarus Fuchs, Georg Frobenius, Wilhelm Killing, Elwin Christoffel, Paul du Bois-Reymond, Paul Bachmann, Leo Königsberger, Franz Mertens, Arthur Schoenflies, Ludwig Stickelberger, Carl Runge, Hans von Mangoldt and Adolf Kneser.

Retirement and legacy

On 23 February 1882 Kummer announced that he would retire, saying that he could no longer trust his memory to do creative work. He retired in 1883, and his student Lazarus Fuchs succeeded him. He lived another decade in Berlin, and died there on 14 May 1893, aged 83.

Legacy

• Algebraic number theory: Dedekind’s ideals (1871) and Kronecker’s divisors were two answers to the question of what Kummer’s ideal numbers really are. Hilbert’s Zahlbericht (1897) organized the subject around Kummer fields, and its last part is devoted to them.
• Fermat and cyclotomy: the regular-prime theorem was the best general result on Fermat’s Last Theorem for more than a century. The Kummer–Vandiver conjecture, that \(p\) never divides the class number of the real cyclotomic field, is still open. Iwasawa theory and the Herbrand–Ribet theorem refine his criterion.
• Class field theory: Kummer theory describes abelian extensions of fields with enough roots of unity and is a building block of class field theory.
• Geometry: Kummer surfaces and K3 surfaces are central in algebraic geometry, string theory and cryptography, where Kummer surfaces give fast arithmetic for genus-2 curves.
• Analysis: Kummer’s function \(M(a,c,z)\), the 24 solutions and Kummer’s test are standard tools.

Connections

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

Teachers and advisors

• Heinrich Ferdinand Scherk 1831

Students

• Georg Cantor 1867
• Georg Frobenius 1870
• Leopold Kronecker
• Adolf Kneser 1884
• Arthur Schoenflies 1877
• Carl Runge 1880
• Elwin Bruno Christoffel 1856
• Ferdinand Joachimsthal
• Franz Mertens 1865
• Gotthold Eisenstein 1845
• Hans von Mangoldt 1878
• Hermann Amandus Schwarz 1864
• Lazarus Fuchs 1858
• Leo Königsberger 1860
• Ludwig Stickelberger 1874
• Paul Bachmann 1862
• Paul du Bois-Reymond 1859
• Wilhelm Killing 1872

Influenced by

• Carl Gustav Jacob Jacobi

Influenced

• Adolf Hurwitz 1877-1878
• Andrew Wiles
• David Hilbert
• Leopold Kronecker 1830s-1891
• Paul Gordan 1855
• Richard Dedekind

Colleagues and collaborators

• Bernhard Riemann
• Carl Gustav Jacob Jacobi
• Jakob Steiner
• Joseph Liouville 1847
• Karl Weierstrass 1856-1883
• Peter Gustav Lejeune Dirichlet

Rivals

• Hermann Grassmann 1847
• Gabriel Lamé 1847

Family

• Hermann Amandus Schwarz

Also mentioned in this life

• André Weil
• Friedrich Wilhelm Bessel
• Leonhard Euler
• Pierre de Fermat
• Augustin-Jean Fresnel
• Évariste Galois
• Carl Friedrich Gauss
• Charles Hermite
• Adrien-Marie Legendre
• August Ferdinand Möbius

Fields

• algebraic number theory 9 lives
• analysis 20 lives
• algebraic geometry 17 lives

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Ernst Eduard Kummer, MacTutor History of Mathematics.
2. Ernst Kummer, Wikipedia.
3. Kummer, Ernst Eduard, Neue Deutsche Biographie and Allgemeine Deutsche Biographie.
4. H.-J. Biermann / Dictionary of Scientific Biography, Kummer, Ernst Eduard.
5. Ernst Eduard Kummer, Mathematics Genealogy Project.
6. K. Hensel, Gedächtnisrede auf Ernst Eduard Kummer, in Kummer, Collected Papers, vol. 1.

Mathematics

1. E. E. Kummer, Collected Papers, ed. A. Weil, 2 vols., Springer, 1975.
2. E. E. Kummer, Über die hypergeometrische Reihe, J. reine angew. Math. 15 (1836).
3. E. E. Kummer, Zur Theorie der complexen Zahlen, and Über die Zerlegung der aus Wurzeln der Einheit gebildeten complexen Zahlen in ihre Primfactoren, J. reine angew. Math. 35 (1847).
4. F. Lemmermeyer, Jacobi and Kummer’s ideal numbers, arXiv:1108.6066.
5. J. J. O’Connor and E. F. Robertson, Fermat’s last theorem, MacTutor.
6. H. M. Edwards, Fermat’s Last Theorem: A Genetic Introduction to Algebraic Number Theory, Springer, 1977.
7. L. C. Washington, Introduction to Cyclotomic Fields, Springer, 2nd ed. 1997.
8. R. W. H. T. Hudson, Kummer’s Quartic Surface, Cambridge University Press, 1905.
9. D. Hilbert, Die Theorie der algebraischen Zahlkörper (Zahlbericht), 1897; English translation Springer, 1998.