Niels Henrik Abel

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

• 4.1 editions from the estate
• 4.2 Later editions with annotations

Life

Origin and study

Abel was the son of Søren Georg Abel (1772-1820), a theologian, temporary deputies and philologists with liberal views, and Anne Marie born 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 ), of Newton, Euler, Lagrange, Laplace, d' Alembert and others gave him to read. From the school still exists a class book with the entry of his teacher Holmboe Abel: " ... that he may be the greatest mathematician in the world, if he lives long enough. "

The family situation worsened when his father, who drank more and more, was released in 1820 and died. Holmboe gave Abel a scholarship so that he could attend the University of Christiania in 1821, but at that time in which it was no training in the natural sciences. He devoted himself, partly supported by professors from their own pocket, the self-study and published in the just founded the first scientific paper in Norway. In 1823 he was able to visit Copenhagen, where he lived with an aunt, dealt with elliptic integrals and his future fiancée Christine Kemp met.

In 1824 he finally received a state scholarship, which enabled him to study abroad. At the same time he published his work on the insolubility of quintic equations by the adjunction of roots, albeit in summary form so that she was almost incomprehensible ( he published an expanded version in Crelle magazine 1826).

Trip abroad 1825-1827

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 of Pure and Applied Mathematics ( also called Crelle Journal ). In his journal Abel published many of his works, and not least by Abel ( and shortly thereafter Jacobi, Steiner and others) acquired Crelle new journal its reputation, which could also exist alongside the then more prestigious French mathematics journals. Abel followed after his Norwegian friends who were mainly trained in geology and mining sciences, to Freiberg in Saxony, where he developed his fundamental work on elliptic functions.

In July 1826, he was in what was then the European center of mathematics in Paris. He handed the Academy his great " Paris Treatise " ( until 1841 published in the Comptes Rendues the Academy ) of what was later called Abelian integrals, an October, she came there temporarily lost but by 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 he was diagnosed with tuberculosis - at that time a death sentence. End of 1826, he left Paris and went back to his friends in Berlin.

Last years in Norway

Crelle offered him in Berlin, the editorship of its journal to, but Abel moved back to Norway (May 1827). His scholarship, however, was not renewed here, and he lived from private lessons, debt and private donations from friends. At the same time he wrote several major works, most of which were published by Crelle in his last year and a half. A treatise on elliptic functions appeared in the Astronomische Nachrichten in Danish Altona. He received a temporary professorship at the university and engineering school in Christiania, but no permanent items, so that his hopes to Berlin directed where Crelle stood up for him.

He spent the end of 1828 with his friends near the Froland iron works in Arendal and worked hard. He met again his fiancée Christine Kemp, who worked there as a governess for friends of Abel's family. When he saw death, he recommended it to a friend, the geologist Baltazar Mathias Keilhau, with whom he had been traveling around Europe. At age 26, he died in 1829 of tuberculosis. Shortly afterwards, a letter from Crelle from Berlin, who announced to him a teaching position. Keilhau and Christine Kemp married next year.

Work

Abel introduced a reformulation of the theory of elliptic integral by, in the theory of elliptic functions, by he used their inverse functions. He also advanced the theory on Riemann surfaces of higher genus ( 's elliptic function ) and introduced Abelian integrals. For this, he generalized the well-known already in the case of elliptic integrals, Euler addition theorem ( Abel's theorem). In this field, he most recently worked in intense competition with Carl Gustav Jacob Jacobi.

He was instrumental in helping introduce more stringent methods in the analysis ( Abel partial summation, his work on the convergence of binomial series, etc.).

In 1824 he proved that a general quintic equation can not be solved by a formula ( " radicals" ) and basic arithmetic operations used only roots. Abel was next to Galois, Abel of studies on the unsolvability of equations ( set of Abel - Ruffini ) generalized (so-called Galois theory ), an important co-founder of group theory. Because of this power is called the commutative groups Abelian groups.

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

Eponyms

After Abel following mathematical structures are named:

• Abelian extension, a Galois field extension with abelian Galois group
• Abelian group, a group for which the commutative law
• Abelian identity, an expression for the Wronskian of two linearly independent homogeneous solutions of a linear differential equation of second order
• Abelian integral, an integral of a rational function in the complex plane,
• Abelian integral equation, a special Volterra integral equation of first kind
• Abelian category, a category that essentially behaves like the category of abelian groups
• Abelian partial summation of a particular forming a sum of products of two numbers in each case
• Abelian projection, a special projection in a von Neumann algebra
• Abelian variety, a complete, coherent Gruppenvarietät

In addition, according to Abel following mathematical sets are named:

• Abelian limit theorem, a set of convergence of a power series in the edge point of the convergence interval
• Abelian Lemma, a set of absolute and locally uniform convergence of power series

Next are named after Abel:

• Abel Prize, a highly doped price the Norwegian Academy of Science awarded annually since 2003
• Abel (crater ), a lunar crater in the northwest of Mare Australe on the lunar front side

Writings

( PDF files at The Works of Niels Henrik Abel )

• Almindelig method til at funktioner find af een variable Störrelse, naar en Egenskab af disse Functioner he udtrykt ved en ligning imellem to variable Magazine for Naturvidenskaberne bind I, 1823, pp. 216-229
• Oplösning af et par Opgaver ved Hjelp af bestemte integral ( del 1) Magazine for Naturvidenskaberne bind II, 1823, pp. 55-68
• Oplösning Nogle af Opgaver ved Hjelp af bestemte integral ( del 2), magazine for Naturvidenskaberne bind II, 1823, pp. 205-215
• Om Maanen Indflydelse paa Pendelens Bevægelse, Magazine for Naturvidenskaberne bind I, 1824, pp. 219-226, Berigtigelse, bind II, 1824, pp. 143-144
• Mémoire sur les équations algébriques où l' on démontre impossibilité de la résolution de l' équation générale du cinquième Degre, Groen Dahl, Christiania 1824
• Det end -celled integral Σnφx udtrykt ved et enkelt bestemt integral magazine for Naturvidenskaberne bind II, 1825, pp. 182-189
• Et Lidet bidrag til lar om adskillige transcendente Functioner, Det Norske Kongelige Videnskabers Selskabs Skrifter 2, 1826, pp. 177-207
• In Journal of Pure and Applied Mathematics ( Crelle Journal ) 1, 1826: Examination of the functions of two independent variables x and y are variable, as f ( x, y), which have the property that f (x, f (x, y )) is a symmetric function of z, x and y, p 11 -15
• Generally resolve proof of the impossibility algebraic equations of higher degrees than the fourth, p 65-84
• Comments on the paper No. 4, page 37 in the first issue of this journal, pp. 117-118
• Resolution of a mechanical task, p 153-157
• Evidence of an expression from which the binomial formula is a single case, p 159-160
• Concerning the integration of the differential formula ρ dx / √ R, when R and ρ integral functions are, pp. 185-221
• Studies on the number: 1 ( M / 1 ), x m × (m-1 ) / ( 1.2) ² · x m × ( m-1) * (M -2 ) / ( 1.2.3 ) · x ³ ...... etc, pp. 311-339

• Recherches sur les fonctions elliptiques, pp. 101-181
• Théorèmes et problèmes, p 286
• Concerning the functions which satisfy the equation phi.x øy = ψ (x y fx fy ) do enough, p 386-394

• Note sur le mémoire de Mr. Olivier L. No.. 4 you second tome de ce journal, ayant pour titre " remarques sur les séries infinies et leur convergence", pp. 79-81
• Recherches sur les fonctions elliptiques. ( Suite du mémoire No. 12 tom. Cah II. 2 de ce journal ), pp. 160-190
• Task from number theory, pp. 212
• Remarques sur quelques propriétés générales d'une certaine varietal de fonctions transcendantes, pp. 313-323
• Sur le nombre of transformations différentes, qu'on peut faire subir à une fonction elliptique par la fonction d'une substitution donné de premier degré, pp. 394-401
• Theorems général sur la transformation des fonctions de la seconde elliptiques et de la troisième espèce, p 402

• Note sur quelques formules elliptiques, pp. 85-93
• Mémoire sur une classe d' équations particulière 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

• Mathematical fragments of Mr NH Abel's letters, Journal of Pure and Applied Mathematics ( Crelle Journal ) 5, 1830, pp. 336-343
• Ferriere mathematical fragments of Mr NH Abel's letters, Journal of Pure and Applied Mathematics ( Crelle Journal ) 6, 1830, pp. 73-80
• Recherches sur les fonctions elliptiques. Second mémoire, Acta Mathematica 26, 1902, pp. 3-41 (in French, with abelprisen.no: , PDF file, 1.3 MB, the preface the editor, PDF file, 83 kB)
• A letter of Niels Henrik Abel, Edmund Jacob Külp, Journal of Pure and Applied Mathematics ( Crelle's Journal ) 125, 1903, pp. 237-240

Later editions with annotations

• Bernt Michael Holmboe (ed.): Oeuvres Completes de NH Abel, mathematicien, avec des notes et développements, rédigées par ordre du roi two volumes, Chr Grondahl, Christiania 1839 (first band, second band )
• Ludwig Sylow, Sophus Lie ( eds.): Oeuvres Completes de Niels Henrik Abel. Nouvelle édition publiée aux frais de l' État Norvégien two volumes, Grøndahl & Son, Christiania 1881
• Hermann Maser ( eds.): Essays on the algebraic solution of equations of NH Abel and E. Galois, Julius Springer, Berlin 1889
• Albert Wangerin (ed.): Studies on the series: 1 ( m / 1) x m · (m- 1) / ( 1.2) * x ² m · (m-1 ) · ( m-2) / ( 1.2.3 ) · x ³ ... ( Crelle Journal 1, 1826, pp. 311-339 ), Wilhelm Engelmann, Leipzig 1895
• Alfred Loewy (ed.): Treatise on a particular class of algebraically solvable equations ( Crelle Journal 4, 1829, pp. 131-156 ), Wilhelm Engelmann, Leipzig 1900