Niels Henrik Abel

from Wikipedia, the free encyclopedia
Niels Henrik Abel.
Gustav Vigeland's Abel sculpture in Oslo .
Memorial plaque on the house, Am Kupfergraben 4a, in Berlin-Mitte

Niels Henrik Abel (born August 5, 1802 on the island of Finnøy , Ryfylke , Norway ; † April 6, 1829 in Froland , Aust-Agder , Norway) was a Norwegian mathematician .


Origin and studies

Abel was the son of Søren Georg Abel (1772–1820), a theologian , intermittent MP and philologist with liberal views, and Anne Marie b. Simonsen (1781-1846). From 1804 he grew up with five siblings in Gjerstad . From 1817 he attended the cathedral school of Christiania (Oslo), where he was strongly encouraged by his teacher Bernt Michael Holmboe (1795-1850), who gave him Newton , Euler , Lagrange , Laplace , d'Alembert and others to read. There is still a class register from his school days with the entry of his teacher Holmboe about Abel: "... that he can become the greatest mathematician in the world if he lives long enough."

The family situation worsened when his father, who drank increasingly, was released and died in 1820. Holmboe gave Abel a scholarship so that he could attend the University of Christiania in 1821, which at the time did not offer any training in the natural sciences. He devoted himself to self-study, partly supported by professors from their own pockets, and published in Norway's first scientific newspaper, which had just been founded. In 1823 he was able to visit Copenhagen , where he stayed with an aunt, worked on elliptical integrals and met his future fiancée Christine Kemp.

In 1824 he finally received a state grant that enabled him to study abroad. At the same time he published his work on the unsolvability of equations of the fifth degree by the adjunction of roots , but in such a concise form that it was almost incomprehensible (he published an expanded version in Crelle's magazine in 1826).

Travel abroad from 1825 to 1827

In 1825 he went to Berlin , where he was supported and promoted by Leopold Crelle , the Berlin engineer, publisher and founder (1826) of the Journal for Pure and Applied Mathematics (also called Crelles Journal ). Abel published many of his work in the latter's journal, and it was through Abel (and shortly afterwards Jacobi , Steiner and others) that Crelle's new journal acquired its reputation, which was able to hold its own alongside the then more respected French mathematics journals. Abel then followed his Norwegian friends, who were mainly trained in geology and mining sciences, to Freiberg in Saxony, where he developed his fundamental work on elliptical functions .

In July 1826 he was in what was then the European center of mathematics, in Paris. In October he submitted his large “Paris Treatise” (published in the Academy's Comptes Rendues in 1841 ) on what was later called Abelian integrals to the Academy, but it was temporarily lost there due to Cauchy and Legendre. Abel believed all his life that she was lost. His stay in Paris was unhappy, he was poor, suffered from depression and was diagnosed with tuberculosis - a death sentence at the time. At the end of 1826 he left Paris and went back to his friends in Berlin.

Last years in Norway

Crelle offered him the editor of his journal in Berlin, but Abel moved back to Norway (May 1827). His scholarship was not renewed here, however, and he lived off private lessons, went into debt, and received private donations from friends. At the same time he wrote several large works in his last year and a half, most of which were published by Crelle. A treatise on elliptical functions appeared in the Astronomische Nachrichten in Altona, Denmark . He received a substitute professorship at the university and engineering school in Christiania, but not a permanent position, so that his hopes were directed to Berlin, where Crelle stood up for him.

At the end of 1828 he spent with his friends near the Froland ironworks in Arendal and worked intensively. He met his fiancée Christine Kemp again, who worked there as a governess for friends of Abel's family. When he saw death coming, he recommended her to a friend, the geologist Baltazar Mathias Keilhau , with whom he had been on a trip to Europe. At the age of 26 he died of pulmonary tuberculosis in 1829. Shortly afterwards a letter arrived from Crelle in Berlin announcing a position as a lecturer. Keilhau and Christine Kemp married the next year.


Abel reformulated the theory of the elliptic integral into the theory of elliptic functions using their inverse functions . He also extended the theory to higher sex Riemann surfaces ( corresponds to the elliptical functions) and introduced Abelian integrals . For this he generalized the addition theorems already known to Euler in the case of elliptic integrals (Abel's theorem). He recently worked in this area in intense competition with Carl Gustav Jacob Jacobi . He also proved that the lemniscate can be divided into n equal parts with a compass and ruler exactly for (where the Fermat prime numbers are).

He was instrumental in introducing more rigorous methods into analysis ( Abelian partial summation , his work on the convergence of binomial series, etc.).

In 1824 he proved that a general fifth-degree equation could not be solved by a formula that only used roots ("radicals") and basic arithmetic operations . Abel was next to Galois , who generalized Abel's investigations into the unsolvability of equations ( theorem of Abel-Ruffini ) (so-called Galois theory ), an important co-founder of group theory . At the beginning of his investigations Abel did not know the work of Paolo Ruffini , but referred to him as his forerunner in a later work from his estate (whose writings, however, are so difficult to read that it is difficult to judge them, but they seem to have gaps ). Because of this achievement, the commutative groups are called Abelian groups .

In 1839 the Norwegian government published his works (edited by his former teacher Holmboe), and in a more complete form in 1881 by his compatriots Peter Ludwig Mejdell Sylow and Sophus Lie .


On April 6, 2014 , a memorial plaque was placed on his former home in Berlin-Mitte , Am Kupfergraben 4a .


The following mathematical structures are named after Abel:

In addition, the following mathematical theorems are named after Abel:

  • Abelian limit theorem , a theorem about the convergence of a power series at the edge point of the convergence interval
  • Abel's criterion , a convergence criterion for infinite series
  • Abel's Lemma , a theorem for the absolute and locally uniform convergence of power series

Also named after Abel:


( PDF files under The Works of Niels Henrik Abel )

  • Almindelig method to find functioners af een variabel Störrelse, naar en Egenskab af disse functioner er udtrykt ved en ligning imellem to variable , Magazin for Naturvidenskaberne bind I, 1823, S. 216-29
  • Oplösning af et Par Opgaver ved Hjelp af bestemte Integraler (del 1), Magazin for Naturvidenskaberne bind II, 1823, pp. 55–68
  • Oplösning af nogle Opgaver ved Hjelp af bestemte Integraler (del 2), Magazin for Naturvidenskaberne bind II, 1823, pp. 205–215
  • Om Maanens Indflydelse paa Pendelens Bevægelse , Magazin for Naturvidenskaberne bind I, 1824, pp. 219–226, Berigtigelse , bind II, 1824, pp. 143–144
  • Mémoire sur les équations algébriques où on démontre l'impossibilité de la résolution de l'équation générale du cinquième dégré , Groendahl, Christiania 1824
  • Det endelige Integral ∑ n φx udtrykt ved et enkelt bestemt Integral , Magazin for Naturvidenskaberne bind II, 1825, pp. 182-189
  • Et lidet Bidrag til Læren om adskillige transcendente Functioner , Det Kongelige Norske Videnskabers Selskabs Skrifter 2, 1826, pp. 177–207
  • in Journal for Pure and Applied Mathematics (Crelles Journal) 1 , 1826:
    • Investigation of the functions of two independently variable quantities x and y, such as f (x, y), which have the property that f (z, f (x, y)) is a symmetric function of z, x and y , p. 11 -15
    • Proof of the impossibility of solving algebraic equations of degrees higher than the fourth in general , pp. 65–84
    • Comments on Paper No. 4, page 37. in the first issue of this journal , pp. 117–118
    • Resolution of a mechanical task , pp. 153–157
    • Proof of an Expression of which the Binomial Formula is a Single Case , pp. 159-160
    • About the integration of the differential formula ρ dx / √R, if R and ρ are whole functions , pp. 185–221
    • Investigations over the series: 1 + (m / 1) x + m · (m − 1) / (1 · 2) · x² + m · (m − 1) · (m − 2) / (1 · 2 · 3 ) · X³ + …… etc. , pp. 311–339
  • in Journal for Pure and Applied Mathematics (Crelles Journal) 2 , 1827:
    • About some definite integrals , pp. 22-30
    • Recherches sur les fonctions elliptiques , pp. 101-181
    • Théorèmes et Genealogie , p. 286
    • About the functions which satisfy the equation φx + φy = ψ (x fy + y fx) , pp. 386–394
  • in Journal for Pure and Applied Mathematics (Crelles Journal) 3 , 1828:
    • Note sur le mémoire de Mr. L. Olivier No. 4. du second tome de ce journal, ayant pour titre “remarques sur les séries infinies et leur convergence” , pp. 79-81
    • Research on les fonctions elliptiques. (Suite du mémoire No. 12. tom. II. Cah. 2. de ce journal) , pp. 160–190
    • Exercise from number theory , p. 212
    • Remarques sur quelques propriétés générales d'une certaine sorte de fonctions transcendantes , pp. 313–323
    • Sur le nombre des transformations différentes, qu'on peut faire subir à une fonction elliptique par la substitution d'une fonction donné de premier degré , pp. 394–401
    • Théorème général sur la transformation des fonctions elliptiques de la seconde et de la troisième espèce , p. 402
  • in Journal for Pure and Applied Mathematics (Crelles Journal) 4 , 1829:
    • Note sur quelques formules elliptiques , pp. 85-93
    • Mémoire sur une classe particulière d'équations résolubles algébriquement , pp. 131–156
    • Théorèmes sur les fonctions elliptiques , pp. 194-199
    • Démonstration d'une propriété générale d'une certaine classe de fonctions transcendentes , pp. 200–201
    • Précis d'une théorie des fonctions elliptiques , pp. 236–277
    • Précis d'une théorie des fonctions elliptiques. (Suite) , pp. 309-348

Editions from the estate

Later editions with notes


  • Øystein Ore : Abel, Niels Henrik . In: Charles Coulston Gillispie (Ed.): Dictionary of Scientific Biography . tape 1 : Pierre Abailard - LS Berg . Charles Scribner's Sons, New York 1970, p. 12-17 .
  • Oystein Ore: Niels Henrik Abel, mathematician extraordinary , Minneapolis 1957
  • Horst Elling, Carl Størmer, Ludvig Sylow, Bjørnstjerne Bjørnson (eds.): Niels Henrik Abel: Mémorial publié à I'occasion du centenaire de sa naissance , Oslo, 1902, archive
  • Arild Stubhaug : A flash of lightning. Niels Henrik Abel and his time. Springer-Verlag, Heidelberg 2003, ISBN 3-540-41879-2
  • Peter Pesic: Abel's proof , Springer-Verlag, Heidelberg 2005, ISBN 3-540-22285-5
  • Magnus Gösta Mittag-Leffler : Niels Henrik Abel. , Revue du Mois, Volume 4, No. 19/20, July 10th / 18th August 1907, pp. 5-25, 207-229. Gutenberg eText (French)
  • August Crelle : Nécrologe , Journal for pure and applied mathematics (Crelles Journal) 4, 1829, pp. 402–404 (obituary; French)
  • Hans Wußing : Niels Henrik Abel , in: H. Wußing, W. Arnold: Biographies of important mathematicians , 3rd edition, Berlin: Volk und Wissen, 1983
  • Marcus du Sautoy The moonlight seekers. Mathematicians unlock the secret of symmetry. CH Beck 2008. ISBN 978-3406576706 .
  • Olav Arnfinn Laudal, Ragni Piene (eds.): The legacy of Niels Henrik Abel: The Abel bicentennial, Oslo, 2002 , Springer 2004
  • Michael Rosen : Niels Henrik Abel and the equation of the fifth degree , Amer. Math. Monthly, Vol. 102, 1995, pp. 495-505. MAA, pdf

Web links

Commons : Niels Henrik Abel  - collection of images, videos and audio files
Wikisource: Niels Henrik Abel  - Sources and full texts

Individual evidence

  1. Michael Rosen, Abel's theorem on the lemniscate, American Mathematical Monthly, Volume 88, 1981, pp. 387-395. In his Disquisitiones Arithmeticae Gauss had proven a corresponding theorem for the circle. In Gauss's diary there is also an entry about the division of the lemniscate into five equal parts with a compass and ruler.
  2. ^ Oystein Ore, Abel article in Dictionary of Scientific Biography