1810 – 1893
Ideal Numbers, Kummer Surfaces & Fermat's Last Theorem for Regular Primes
Ernst Eduard Kummer was born on 29 January 1810 in Sorau, Brandenburg (now Zary, Poland). His father, a physician, died of typhus during the Napoleonic campaigns when Ernst was only three years old.
Despite financial hardship, Kummer's exceptional mathematical talent was recognised early. He attended the Gymnasium in Sorau, where his teachers noted his extraordinary facility with numbers and abstract reasoning.
In 1828, he entered the University of Halle to study Protestant theology, but quickly shifted to mathematics under the influence of Heinrich Ferdinand Scherk. His doctoral dissertation (1831) concerned a series expansion related to the hypergeometric function.
Born into modest circumstances in the Prussian province of Brandenburg. Lost his father at age 3.
University of Halle, 1828–1831. Switched from theology to mathematics. PhD on hypergeometric series.
Taught at the Gymnasium in Liegnitz (1832–1842), where one of his pupils was Leopold Kronecker.
Kummer spent a decade as a Gymnasium teacher before Dirichlet recommended him for a professorship at the University of Breslau in 1842.
In 1855, he succeeded Dirichlet at the University of Berlin when Dirichlet moved to Gottingen to replace Gauss. At Berlin, Kummer joined Weierstrass and Kronecker to form what became the most influential mathematical triumvirate in Germany.
He was elected to the Berlin Academy of Sciences and served as its secretary. Kummer also reformed mathematics education at the Berlin War College and supervised numerous doctoral students who became leading mathematicians.
Appointed professor at Breslau on Dirichlet's recommendation.
Succeeds Dirichlet at Berlin. Joins Weierstrass and Kronecker.
Awarded the Grand Prix by the Paris Academy for work on Fermat's Last Theorem.
Retires from Berlin. His legacy shapes algebraic number theory for generations.
Kummer worked during the golden age of German mathematics, when the foundations of modern algebra and number theory were being laid.
Prussia's rise as an intellectual power. The German university system — with its research seminars — became the global model for mathematical education.
Gauss had opened number theory; Dirichlet applied analysis to it; Kummer extended it into algebraic domains. The factorisation problem in cyclotomic fields was the central puzzle.
Fermat's Last Theorem remained unsolved since 1637. The French Academy offered repeated prizes, creating intense international competition that drove Kummer's work.
The concept of "number" was expanding — from integers to Gaussian integers to algebraic integers. The question of which properties survived this expansion was critical.
Kummer, Weierstrass, and Kronecker at Berlin created a powerhouse. Their differing philosophies — constructivist vs. analytical — produced creative tension.
Gauss's Disquisitiones Arithmeticae (1801) set the agenda. Kummer's ideal numbers were the most profound extension of Gauss's vision of arithmetic in higher domains.
In rings of cyclotomic integers, unique factorisation fails. Kummer discovered this in the 1840s while attempting to prove Fermat's Last Theorem.
His revolutionary solution: introduce "ideal numbers" — auxiliary elements that restore unique factorisation in these extended number systems.
For example, in Z[√-5], the number 6 factors as both 2 × 3 and (1+√-5)(1-√-5). Kummer's ideal numbers resolve this ambiguity.
Kummer introduced ideal numbers in his landmark 1847 paper on cyclotomic fields. The key setting was the ring of integers in Q(ζ_p), where ζ_p is a primitive p-th root of unity.
He showed that unique factorisation fails in these rings for certain primes p (the irregular primes). For regular primes — those not dividing the class number of the cyclotomic field — unique factorisation into ideal primes holds.
Kummer's ideal numbers were later reformulated by Dedekind as ideals — subsets of a ring closed under addition and multiplication by ring elements. This reformulation became standard and is fundamental to modern commutative algebra.
The concept of the class number — measuring how far a ring deviates from unique factorisation — became a central invariant in algebraic number theory.
A prime p is regular if it does not divide the class number of Q(ζ_p). The first irregular prime is 37. Kummer proved FLT for all regular primes.
37, 59, 67, 101, 103, 131, 149, ... For these primes, unique factorisation fails in the cyclotomic ring and additional analysis is needed.
Dedekind replaced Kummer's "ideal numbers" with ideals as subsets of rings, providing a cleaner framework that generalised to all algebraic number fields.
Kummer connected the class number to Bernoulli numbers: p is irregular iff p divides the numerator of some B_k for even k < p.
Kummer's most celebrated achievement was his 1850 proof that Fermat's Last Theorem holds for all regular prime exponents.
The equation x^p + y^p = z^p has no non-trivial integer solutions when p is a regular prime. This covered infinitely many cases of FLT at once — a spectacular advance.
His method factored the left side in the cyclotomic field Q(ζ_p) and used the unique factorisation of ideals to derive a contradiction.
The Paris Academy awarded him the Grand Prix des Sciences Mathematiques in 1857, recognising his proof as the most important advance on FLT since Fermat.
Kummer's proof divided into two cases. Case I: p does not divide any of x, y, z. Case II: p divides exactly one of them. Both required deep analysis of units and ideal classes in cyclotomic fields.
He showed that the first case follows from the regularity condition via a descent argument. The second case required his theory of p-adic units in cyclotomic fields — work that anticipated much of modern algebraic number theory.
Kummer also developed computational methods for determining regularity. He computed class numbers for primes up to 163 and showed that among primes less than 100, only 37, 59, and 67 are irregular.
Kummer conjectured that about e^(-1/2) ≈ 60.65% of primes are regular. Computations up to large bounds support this estimate remarkably well.
Kummer's methods covered infinitely many exponents at once. Before him, only individual cases (n=3 by Euler, n=5 by Legendre/Dirichlet, n=7 by Lame) were known.
Kummer's ideal-theoretic approach inspired 150 years of development in algebraic number theory, ultimately contributing to the framework that enabled Andrew Wiles' 1995 proof.
In algebraic geometry, a Kummer surface is a quartic surface in projective 3-space with exactly 16 ordinary double points — the maximum possible for a quartic.
Kummer studied these surfaces in the 1860s, connecting them to the theory of theta functions and abelian varieties. A Kummer surface arises as the quotient of an abelian surface by the involution sending a point to its negative.
The 16 singular points correspond to the 2-torsion points of the abelian surface. These surfaces exhibit a remarkable (16,6) configuration: 16 points and 16 planes, each plane containing 6 points and each point lying on 6 planes.
Kummer surfaces remain important in modern algebraic geometry, K3 surface theory, and string theory compactifications.
"Kummer's method was to enlarge the number system until unique factorisation was restored — a stroke of genius that transformed number theory."
— Harold Edwards, on Kummer's approachFind where unique factorisation fails in cyclotomic rings
Introduce "ideal" divisors to complete the factorisation
Calculate class numbers via Bernoulli numbers
Use restored UF to prove impossibility theorems
Kummer combined algebraic creativity with heavy computation. He was willing to perform extensive numerical calculations — computing class numbers, verifying regularity conditions, tabulating Bernoulli numbers — while maintaining a deep theoretical vision. His approach balanced structural insight with concrete verification, a method that remains the gold standard in algebraic number theory.
Kummer's work sat at the intersection of number theory, algebra, and geometry. His students and intellectual heirs — Kronecker, Dedekind, Hilbert — shaped the foundations of modern algebra.
Kummer's journey to ideal numbers began with failure. In 1843, he believed he had proved Fermat's Last Theorem completely by factoring in cyclotomic fields.
He sent his proof to Dirichlet, who identified the fatal flaw: Kummer had assumed unique factorisation holds in all cyclotomic integer rings. It does not.
Rather than abandoning the approach, Kummer spent the next four years understanding exactly how unique factorisation fails — and then invented ideal numbers to restore it.
This is one of the great examples in mathematics of a productive failure: the wrong assumption led directly to a profound new theory that proved far more important than the original goal.
"The concept of ideal numbers...arose from the attempt to prove higher reciprocity laws and Fermat's theorem."
— Ernst KummerAssumed unique factorisation in Z[ζ_p] for all primes p. Dirichlet showed this fails for p = 23 and beyond.
Instead of a complete FLT proof, Kummer created ideal theory — arguably more valuable than any single theorem it could prove.
Kummer's ideal numbers became Dedekind ideals, the foundation of commutative algebra and algebraic number theory. Every modern textbook traces its lineage to Kummer.
Kummer's class number computations inspired Hilbert's class field theory programme, which Artin, Takagi, and others completed in the 20th century.
The Kummer sequence in algebraic K-theory and etale cohomology bears his name, connecting his cyclotomic work to modern homotopy theory.
Kummer surfaces are a key class of K3 surfaces. Their 16-point configurations appear in string theory compactifications and mirror symmetry.
Kummer's approach motivated 150 years of work that culminated in Wiles' proof (1995) via modularity of elliptic curves — a path Kummer could not have foreseen but helped open.
Kummer's explicit computations with Bernoulli numbers and class numbers established the tradition of computational algebraic number theory.
Modern public-key cryptography (RSA, elliptic curve methods) relies on the arithmetic of algebraic number fields that Kummer pioneered. Ideal theory underpins the security analysis of these systems.
Algebraic number theory, descending from Kummer's work, provides constructions for efficient error-correcting codes used in digital communications and storage.
Kummer surfaces appear in Calabi-Yau compactifications of string theory. The 16 singular points correspond to physical moduli of the compactified dimensions.
Post-quantum cryptography uses ideal lattices in cyclotomic fields — exactly the structures Kummer studied. His ideal theory provides the mathematical framework.
Kummer's abstract investigations into the arithmetic of cyclotomic fields now underpin some of the most practical technologies in digital security and communications.
Harold M. Edwards (1977). The definitive account of Kummer's work on FLT and ideal numbers, reconstructing his arguments in modern language. Essential reading.
Jurgen Neukirch (1999). A comprehensive modern treatment that builds on Kummer's and Dedekind's foundations. The standard graduate text.
Simon Singh (1997). A popular account of the history of FLT from Fermat through Kummer to Wiles. Accessible to general readers.
Kenneth Ireland & Michael Rosen (1990). Covers cyclotomic fields and Kummer's contributions in a rigorous but accessible style.
David Cox (1989). Traces the development from Fermat through Kummer to class field theory, showing how ideal theory evolved.
Andre Weil (1984). A masterful historical survey from Hammurapi to Legendre, with deep analysis of the pre-Kummer context.
"The introduction of ideal numbers by Kummer is one of the finest achievements of the human mind in abstract mathematics."
— Leopold Kronecker, Kummer's student and colleagueErnst Eduard Kummer
1810 – 1893
From a failed proof of Fermat's Last Theorem arose one of the most important ideas in all of algebra.