Niels Henrik Abel

How to use this course

Niels Henrik Abel worked for about eight years. In that time he settled a question about the quintic equation that had been open for nearly three centuries, helped set analysis on rigorous foundations, turned elliptic integrals into elliptic functions, and found the general addition theorem from which the modern theory of algebraic curves and their Jacobians grew. Commutative groups are called abelian after him, and since 2003 the most prestigious prize in mathematics outside the Fields Medal has carried his name.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Lessons 04 and 05 cover his analysis, Lessons 06 and 07 his algebra, Lessons 08 and 09 his transcendental work, and Lesson 10 his earliest paper of lasting importance.

Timeline

Gjerstad and Christiania, 1802–1821

Niels Henrik Abel was born on 5 August 1802 in the southwest of Norway, then united with Denmark. The parish records point to Nedstrand, near Stavanger, and he was baptised in the church at Finnøy. His father, Søren Georg Abel, was a Lutheran pastor with theological training and a strong interest in national politics. His mother, Anne Marie Simonsen, came from a merchant family. In 1804 the family moved to Gjerstad on the southeast coast, where his father took over the parish.

The father taught his sons at home. In 1815 Niels and his elder brother Hans Mathias were sent to the Cathedral School in Christiania, the capital, now Oslo. The school had lost many of its best teachers to the new university, founded in 1811, and Abel’s early record there was unremarkable. Teaching was harsh, and the mathematics teacher was dismissed for brutality toward a pupil.

Holmboe

His replacement was Bernt Michael Holmboe, only seven years older than Abel. Holmboe set original problems, saw at once that Abel was exceptional, and gave him private lessons. Within a year or two Abel was reading Leonhard Euler, Joseph-Louis Lagrange, Pierre-Simon Laplace and Carl Friedrich Gauss’s Disquisitiones arithmeticae, and finding gaps in their arguments. Holmboe wrote in the school report that Abel could become the greatest mathematician in the world if he lived.

In 1820 his father died. The family was left with almost nothing, and Abel, still a schoolboy, took on part of the burden of supporting his mother and younger siblings. Holmboe and other teachers raised money from their own pockets so that he could continue his studies.

Student, Degen and the quintic, 1821–1825

In his last year at school Abel believed he had found a formula in radicals for the roots of the general quintic equation. No one in Norway could judge it, so Holmboe and the astronomer Christopher Hansteen sent it to Ferdinand Degen, the leading mathematician in Copenhagen, for the Royal Danish Academy. Degen could not find an error but asked for a numerical example. Working one out, Abel discovered that his method failed. Degen also advised him to turn to elliptic integrals, which he described as a subject where a strong mind could find new results. It was good advice.

Abel entered the university in Christiania in 1821 with free lodging and support from the professors. He passed the preliminary examination in 1822 and then worked on his own; there was no one to teach him further. Hansteen let him live in his house and printed his first papers in the Magazin for Naturvidenskaberne. One of these, in Norwegian, solved a mechanical problem by what is now called Abel’s integral equation (Lesson 10).

In the summer of 1823 a gift of 100 speciedaler from Professor Søren Rasmussen took him to Copenhagen. He met Degen and other Danish mathematicians, and at a ball he met Christine Kemp, a young woman with no money of her own. They became engaged at Christmas 1824.

The pamphlet of 1824

Back in Christiania, Abel turned the failed solution around and proved that no formula in radicals exists for the general equation of degree five. To save printing costs he compressed the proof into six pages and published it in French at his own expense as Mémoire sur les équations algébriques où l’on démontre l’impossibilité de la résolution de l’équation générale du cinquième degré. He sent copies abroad, including one to Gauss. Gauss never read it; the pamphlet was found uncut among his papers after his death. Abel heard that Gauss was dismissive, and later avoided Göttingen.

Berlin and Paris, 1825–1827

In 1825 the Norwegian government gave Abel a two-year travel grant. He left in September with a group of young Norwegian scientists. In Copenhagen he learned that Degen had died. He went on to Berlin, where he called on August Leopold Crelle, an engineer and amateur mathematician with influence in the Prussian ministry.

Crelle could not follow all of Abel’s work, but he recognized its quality and became his closest friend and advocate. He was founding a new journal, the Journal für die reine und angewandte Mathematik, and Abel’s papers filled much of its early volumes. The first volume, 1826, carried seven of his articles, among them an expanded proof of the impossibility of the quintic and the paper on the binomial series (Lesson 04). In letters home Abel complained about the lack of rigour in the analysis of his day, especially the casual handling of divergent series.

Paris

After a journey through Freiberg, Dresden, Prague, Vienna, northern Italy and Switzerland, Abel reached Paris in July 1826. He hoped the French mathematicians would recognize him. They received him politely and paid little attention. On 30 October 1826 he presented to the Academy of Sciences a long memoir on a very general class of transcendental functions, containing what is now called Abel’s theorem (Lesson 09). Augustin-Louis Cauchy and Adrien-Marie Legendre were appointed to report on it. Legendre found it hard to read; Cauchy put it aside. No report appeared.

Short of money and unwell, Abel left Paris at the end of 1826 and spent some weeks in Berlin. Crelle tried to keep him there, but Abel returned to Christiania in May 1827, heavily in debt and with no post.

The binomial series and rigour

For the general reader. In the early nineteenth century mathematicians manipulated infinite sums as if they were finite. The results were often right and sometimes nonsense. Abel’s 1826 paper on the binomial series was one of the first to say exactly when an infinite series has a sum, and to prove a famous formula with full care for every complex exponent.

Abel’s paper, Untersuchungen über die Reihe \(1+\frac m1x+\frac{m(m-1)}{1\cdot2}x^2+\cdots\) (Crelle’s Journal 1, 1826), treats the series

for complex \(m\) and \(x\). Euler and Lagrange had used it freely; Cauchy had treated real exponents. Abel set out to determine where it converges and what it converges to.

Proof sketch

Absolute convergence for \(|x|<1\) follows from the ratio test, since \(\big|\binom m{k+1}\big/\binom mk\big|=|m-k|/(k+1)\to1\). Multiplying two absolutely convergent series and using Vandermonde’s identity \(\sum_j\binom mj\binom n{k-j}=\binom{m+n}k\) gives the functional equation

Abel proved that \(\phi\) is continuous in \(m\). A continuous solution of \(f(m)f(n)=f(m+n)\) with \(f(1)=1+x\) must be \(f(m)=\exp(cm)\) with \(e^c=1+x\), and continuity in \(x\) from \(x=0\) fixes \(c=\log(1+x)\), the principal branch. For the boundary he used the asymptotics \(\big|\binom mk\big|\approx C\,k^{-\operatorname{Re}m-1}\) together with the summation by parts of Lesson 05.

The exception to Cauchy’s theorem

In his Cours d’analyse of 1821 Cauchy had stated that a convergent series of continuous functions has a continuous sum. In a footnote to the binomial paper Abel remarked politely that this theorem seems to admit exceptions, and gave the series

which converges for every real \(\varphi\) but jumps at \(\varphi=(2k+1)\pi\), where every term vanishes. The missing hypothesis is uniform convergence, isolated later by Philipp Ludwig von Seidel, George Stokes and Karl Weierstrass. Abel’s own proof of continuity in \(m\) in fact used a uniform estimate.

Summation by parts and Abel’s limit theorem

For the general reader. Integration by parts trades a hard integral for an easier one. Abel found the discrete version for sums. With it he proved that a power series which converges at the edge of its disc is continuous up to that edge, a tool used to this day in analysis, number theory and probability.

Summation by parts

Let \(A_n=a_1+\cdots+a_n\) and \(A_0=0\). Writing \(a_n=A_n-A_{n-1}\) and regrouping,

Indeed the right side of the summation formula is bounded by \(Mb_N+M\sum(b_n-b_{n+1})=Mb_1\). Two convergence tests follow. Dirichlet’s test: if \(A_n\) is bounded and \(b_n\downarrow0\), then \(\sum a_nb_n\) converges. Abel’s test: if \(\sum a_n\) converges and \(b_n\) is monotone and bounded, then \(\sum a_nb_n\) converges.

Proof

Replacing \(c_0\) by \(c_0-s\), assume \(s=0\). Let \(s_n=c_0+\cdots+c_n\), so \(s_n\to0\). Summation by parts gives, for \(|x|<1\),

Given \(\varepsilon>0\), choose \(N\) with \(|s_n|<\varepsilon\) for \(n>N\). For \(0<x<1\), \(|f(x)|\le(1-x)\sum_{n\le N}|s_n|+\varepsilon(1-x)\sum_{n>N}x^n\le(1-x)\sum_{n\le N}|s_n|+\varepsilon\). The first term tends to 0 as \(x\to1\). In a Stolz angle the same argument works with \(\varepsilon\,|1-z|/(1-|z|)\le K\varepsilon\).

Abel summation

The converse fails: \(\lim_{x\to1^-}\sum c_nx^n\) may exist when \(\sum c_n\) diverges. One then calls the limit the Abel sum of the series. Abel himself distrusted divergent series, but Abel summation became a central method of the theory developed by G. H. Hardy, and its partial converses, the Tauberian theorems, are a subject of their own.

The impossibility of the quintic

For the general reader. Babylonian scribes could solve quadratic equations, and Italian algebraists of the sixteenth century found formulas for cubics and quartics using square and cube roots. For 250 years mathematicians searched for a formula for degree five. Abel proved there is none: no expression built from the coefficients by arithmetic and root extraction can give the roots of the general quintic.

Paolo Ruffini had published an argument in 1799 and refined it later. It assumed, without proof, that every radical in a solution formula can be written as a rational function of the roots. Abel proved this lemma, now called his theorem on natural irrationalities, and with it gave the first complete proof. The 1824 pamphlet was very compressed; the 1826 version in Crelle’s Journal is fuller. Pierre Wantzel published a simplified proof in 1845.

Proof sketch

Write the general polynomial as \(\prod(x-x_i)\) with \(x_1,\dots,x_n\) independent indeterminates, so the coefficients are the elementary symmetric functions. Adjoin all roots of unity to the ground field at the start; this does not affect the argument. The symmetric group \(S_n\) permutes the \(x_i\).

1. Natural irrationalities. If the roots lie in a radical tower, then one can choose a tower whose every element is a rational function of \(x_1,\dots,x_n\). So each stage is a field \(K_j\subseteq k(x_1,\dots,x_n)\) with \(K_{j+1}=K_j(r_j)\), \(r_j^{p_j}\in K_j\), \(p_j\) prime.
2. Commutator lemma. Let \(H\) be a set of permutations fixing \(K_j\) elementwise. For \(\sigma\in H\), \(\sigma(r_j)^{p}=r_j^{p}\), so \(\sigma(r_j)=\zeta_\sigma r_j\) for a root of unity \(\zeta_\sigma\). Hence every commutator \(\sigma\tau\sigma^{-1}\tau^{-1}\) with \(\sigma,\tau\in H\) fixes \(r_j\), and therefore fixes \(K_{j+1}\).
3. 3-cycles. All permutations fix \(K_0\). For \(n\ge5\), every 3-cycle is a commutator of two 3-cycles: \((a\,b\,c)=[(a\,b\,d),(a\,c\,e)]\) up to relabelling, using five distinct letters. By induction, every 3-cycle fixes every \(K_j\).
4. Contradiction. The 3-cycle \((1\,2\,3)\) moves \(x_1\), so \(x_1\) lies in no \(K_j\).

Step 3 is where \(n\ge5\) enters: with four letters there is no room for the two auxiliary letters, and indeed the quartic is solvable. In Galois’s language, \(S_n\) is not solvable for \(n\ge5\).

Abelian equations and equations of prime degree

For the general reader. Having shown that some equations cannot be solved by radicals, Abel asked which ones can. He found a large class that always can, defined by a commutativity condition among their roots. That condition is why commutative groups are now called abelian.

In modern terms the hypotheses say that \(K(\alpha)\) is the splitting field of \(f\) and that its Galois group, whose elements are \(\alpha\mapsto\theta_i(\alpha)\), is commutative. A finite abelian group is a product of cyclic groups, and a cyclic extension of prime degree \(p\) over a field containing the \(p\)-th roots of unity is generated by a \(p\)-th root (Lagrange resolvents). Abel argued along these lines, and the theorem appeared in Mémoire sur une classe particulière d’équations résolubles algébriquement, Crelle’s Journal 4 (1829). Leopold Kronecker later called such equations abelian, and the name passed to commutative groups.

Lagrange resolvents

Suppose \(L/K\) is cyclic of prime degree \(p\), generated by \(\sigma\), and \(\zeta\in K\) is a primitive \(p\)-th root of unity. For \(\beta\in L\) put \(\rho=\beta+\zeta\sigma\beta+\cdots+\zeta^{p-1}\sigma^{p-1}\beta\). Then \(\sigma\rho=\zeta^{-1}\rho\), so \(\rho^p\) is fixed by \(\sigma\) and lies in \(K\). Choosing \(\beta\) with \(\rho\ne0\) gives \(L=K(\rho)\) with \(\rho^p\in K\).

Abel announced this in a letter to Crelle of 18 October 1828. A proof was published by Évariste Galois, who showed that the Galois group must then be a subgroup of the affine group \(x\mapsto ax+b\) over \(\mathbb Z/p\), in which only the identity fixes two points.

Elliptic functions by inversion

For the general reader. The arc length of a circle leads to the integral \(\int dx/\sqrt{1-x^2}=\arcsin x\), and the sine function is its inverse. Arc lengths of ellipses and other curves give integrals with square roots of cubic or quartic polynomials. Legendre had spent forty years studying these elliptic integrals. Abel’s idea was to invert them, as sine inverts arcsine. The inverse functions turned out to have two independent periods, and the subject was transformed.

In Recherches sur les fonctions elliptiques (Crelle’s Journal 2 and 3, 1827–28) Abel defined \(\varphi(\alpha)=x\) by

extended \(\varphi\) to complex arguments, and showed it is doubly periodic and meromorphic, with periods \(2\omega\) and \(2\tilde\omega i\), where \(\omega/2=\int_0^{1/c}\) and \(\tilde\omega/2=\int_0^{1/e}\) of the corresponding real integrands. Jacobi’s function \(\operatorname{sn}\) is the same object in a different normalization. The simplest case \(c=e=1\) is the lemniscatic sine.

The lemniscatic sine

Define \(\operatorname{sl}\) on \([-\varpi/2,\varpi/2]\) by \(u=\int_0^{\operatorname{sl}u}\frac{dt}{\sqrt{1-t^4}}\), where

Then \((\operatorname{sl}'u)^2=1-\operatorname{sl}^4u\). The substitution \(t\mapsto it\) shows \(\operatorname{sl}(iu)=i\operatorname{sl}u\), so the real period \(2\varpi\) is accompanied by the imaginary period \(2i\varpi\).

The addition formula is Euler’s addition theorem for the lemniscatic integral, read in the inverse direction. In Abel’s hands it does the work of a definition: it extends \(\operatorname{sl}\) from a small interval to the whole plane, since \(\operatorname{sl}(u+iv)\) is expressed through real values of \(\operatorname{sl}\) and \(\operatorname{sl}'\).

Division of the lemniscate

The arc of the lemniscate \(r^2=\cos2\theta\) from the origin to a point at distance \(r\) has length \(\int_0^r dt/\sqrt{1-t^4}\), so the whole curve has length \(2\varpi\). Dividing it into \(n\) equal arcs amounts to computing \(\operatorname{sl}(2\varpi k/n)\). Gauss had hinted in the Disquisitiones that his theory of the circle extended to the lemniscate. Abel proved that the division is possible with ruler and compass whenever \(n\) is a power of 2 times a product of distinct Fermat primes, exactly as for the circle. The converse was proved by Michael Rosen in 1981. The proof uses complex multiplication: \(\operatorname{sl}((1+i)u)\) is a rational function of \(\operatorname{sl}u\), so the division equations are abelian over \(\mathbb Q(i)\), and Lesson 07 applies.

Abel’s theorem and the Paris memoir

For the general reader. Euler had found that the sum of two elliptic integrals is another elliptic integral, with an algebraic correction. Abel generalized this to every integral of an algebraic function. A sum of any number of such integrals can be reduced to a fixed number of them, and that fixed number, depending only on the underlying curve, is what Bernhard Riemann later called the genus. This is the theorem Abel submitted in Paris in 1826.

Abel’s method on a cubic

Let \(E:\ y^2=x^3+ax+b\) and \(\omega=dx/y\). A line \(y=mx+c\) meets \(E\) at three points \((x_i,y_i)\), the roots of \(F(x)=x^3+ax+b-(mx+c)^2\). Let \(m,c\) vary. Differentiating \(F(x_i)=0\),

For a monic polynomial \(F\) of degree 3 with distinct roots, \(\sum_i x_i^k/F'(x_i)=0\) for \(k=0,1\) (compare coefficients in the partial fraction expansion of \(x^k/F(x)\)). Hence \(\sum_i\omega(P_i)\) is identically zero, and \(\sum_i\int_{O}^{P_i}\omega\) is constant modulo periods as the line moves. This is the group law on \(E\): three collinear points sum to zero.

Abel’s original formulation was more general and more computational. For an algebraic relation \(\chi(x,y)=0\) and any integral \(\int R(x,y)\,dx\), he showed that the sum over the intersection points of the curve with a varying auxiliary curve is an algebraic-logarithmic function of the parameters. He also showed that any sum of such integrals reduces to a fixed number of them, and computed that number, the genus, from the curve. With Jacobi’s inversion problem, solved for hyperelliptic curves by Göpel and Rosenhain and in general by Riemann and Weierstrass, this gives the Jacobian variety \(\mathbb C^g/\Lambda\).

The memoir’s fate

Abel presented the memoir, Mémoire sur une propriété générale d’une classe très étendue de fonctions transcendantes, on 30 October 1826. After Abel’s death, prompted by enquiries about it, Cauchy found the manuscript in 1830. It was printed in 1841 in the Mémoires présentés par divers savants. The manuscript then vanished again, was located in Florence in 1952, and was shown in 2002 by Andrea Del Centina to survive there in its entirety.

Abel’s integral equation

For the general reader. A bead slides without friction down a wire. If you know how long it takes to reach the bottom from every starting height, can you recover the shape of the wire? Abel solved this inverse problem in 1823. His solution is the first solution of an integral equation and an early instance of fractional calculus.

A bead released at rest at height \(h\) on a curve with arc length \(s(y)\) measured from the bottom reaches speed \(\sqrt{2g(h-y)}\) at height \(y\). Its descent time is

Given \(T\), find \(s'\). Abel posed the general form in the 1823 Norwegian paper and returned to it in Auflösung einer mechanischen Aufgabe, Crelle’s Journal 1 (1826).

Proof

Integrate the equation against \((y-x)^{\alpha-1}\) and exchange the order of integration:

The inner integral is independent of \(t\) after the substitution \(x=t+(y-t)u\), and \(B(\alpha,1-\alpha)=\Gamma(\alpha)\Gamma(1-\alpha)=\pi/\sin\pi\alpha\). Differentiate in \(y\).

The tautochrone

Suppose \(T(h)=T\) is constant. With \(\alpha=\tfrac12\), \(\varphi=s'\) and \(f=\sqrt{2g}\,T\), the formula gives \(s'(y)=\frac{\sqrt{2g}\,T}{\pi}\frac{d}{dy}\big(2\sqrt y\big)=\frac{\sqrt{2g}\,T}{\pi\sqrt y}\). A curve with \(ds/dy=\sqrt{2r/y}\) is a cycloid generated by a circle of radius \(r\), and then \(T=\pi\sqrt{r/g}\). Christiaan Huygens had found this tautochrone property in 1659; Abel’s method shows the cycloid is the only such curve. (The constant-\(T\) case does not satisfy \(f(0)=0\); the formula still applies in the sense of distributions, or directly by checking the answer.)

The race with Jacobi and the last winter, 1827–1829

Back in Christiania, Abel had no position. He tutored schoolboys and lived on small loans and a modest university stipend. Christine Kemp took work as a governess at the Froland ironworks, near Arendal. In 1828, when Hansteen left on a long magnetic survey expedition to Siberia, Abel was appointed to teach in his place at the university and at the Norwegian Military Academy. His situation improved a little.

The first part of the Recherches appeared in 1827. That year Carl Gustav Jacobi, then at Königsberg, published short notes in the Astronomische Nachrichten announcing results on the transformation of elliptic integrals, without proofs. Abel saw quickly that they followed from his own theory, and added a section proving them. For about a year the two young men published in rapid succession, Abel in Crelle’s Journal and Jacobi in the Fundamenta nova theoriae functionum ellipticarum (1829). Legendre, in his seventies, wrote to Jacobi that the two of them would stand among the foremost analysts of the age.

Abel spent the summer of 1828 at Froland. Crelle and others pressed the Prussian authorities to create a post for him in Berlin. At Christmas 1828 Abel travelled by sled through severe cold to visit his fiancée at Froland. He fell ill there with tuberculosis, which had probably affected him since Paris. He continued to write; a short paper of early January 1829 sketched the general theorem of the Paris memoir again, in case the manuscript was lost. He died at Froland on 6 April 1829, aged 26. On 8 April Crelle wrote with the news that he had been appointed to a professorship in Berlin.

Recognition and legacy

In 1830 the Paris Academy awarded its Grand Prix in mathematics jointly to Abel, posthumously, and to Jacobi. Christine Kemp later married Abel’s friend, the geologist Baltazar Mathias Keilhau. Holmboe edited the first edition of Abel’s works in 1839. A second, fuller edition by Ludwig Sylow and Sophus Lie, two Norwegian mathematicians whose own work built on Abel’s, appeared in 1881.

Influence

• Algebra: Abel’s question of which equations are solvable was answered completely by Galois. The adjective abelian, for commutative groups, became standard by the end of the nineteenth century, and abelian extensions are the subject of class field theory.
• Analysis: his insistence on convergence, his summation by parts and his limit theorem are part of every course in analysis.
• Elliptic and abelian functions: the inversion of elliptic integrals launched the theory developed by Jacobi, Weierstrass and Riemann. Abel’s theorem, the genus and the Jacobian are the origin of the algebraic geometry of curves, and abelian varieties are named for him.
• Integral equations: the Abel transform is used in tomography, plasma diagnostics and seismology.

Honours

In 1899 Sophus Lie, learning that Alfred Nobel’s prizes would not include mathematics, proposed an Abel Prize for the centenary of Abel’s birth. King Oscar II offered funding in 1902, but the plan lapsed after the dissolution of the union between Norway and Sweden in 1905. The Norwegian government finally established the Abel Prize in 2001 for the bicentenary; Atle Selberg received an honorary prize in 2002, and Jean-Pierre Serre the first regular prize in 2003. Abel is commemorated by Gustav Vigeland’s monument in Oslo, by the lunar crater Abel, and on Norwegian stamps and banknotes.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Niels Henrik Abel, MacTutor History of Mathematics.
2. Niels Henrik Abel, Wikipedia.
3. A. Stubhaug, Niels Henrik Abel and his Times: Called Too Soon by Flames Afar, Springer, 2000.
4. O. Ore, Niels Henrik Abel: Mathematician Extraordinary, University of Minnesota Press, 1957.
5. Abel Prize, Wikipedia (history of the prize).
6. A. Del Centina, The manuscript of Abel’s Parisian memoir found in its entirety, Historia Mathematica 29 (2002) 65–69.

Mathematics

1. N. H. Abel, Œuvres complètes, ed. L. Sylow and S. Lie, 2 vols., Christiania, 1881.
2. N. H. Abel, Beweis der Unmöglichkeit, algebraische Gleichungen von höheren Graden als dem vierten allgemein aufzulösen, J. reine angew. Math. 1 (1826).
3. N. H. Abel, Untersuchungen über die Reihe …, J. reine angew. Math. 1 (1826) 311–339.
4. N. H. Abel, Recherches sur les fonctions elliptiques, J. reine angew. Math. 2 (1827), 3 (1828).
5. N. H. Abel, Mémoire sur une classe particulière d’équations résolubles algébriquement, J. reine angew. Math. 4 (1829).
6. N. H. Abel, Mémoire sur une propriété générale d’une classe très étendue de fonctions transcendantes, Mém. prés. par divers savants 7 (1841). Translation and commentary by J. Little.
7. P. Pesic, Abel’s Proof, MIT Press, 2003.
8. M. Rosen, Abel’s theorem on the lemniscate, Amer. Math. Monthly 88 (1981) 387–395.
9. Abel–Ruffini theorem and Abel–Jacobi map, Wikipedia.
10. P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley, 1978, chapter 2 (Abel’s theorem).