Gotthold Eisenstein

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Gotthold Eisenstein

Ferdinand Gotthold Max Eisenstein (born April 16, 1823 in Berlin ; † October 11, 1852 there ) was a German mathematician who mainly worked in number theory and on elliptic functions .

Origin and youth

He was the son of the merchant and temporary plating manufacturer Johann Konstantin Eisenstein (1791–1875), born in Danzig, and of Helene Pollack (1799–1876), originally from Königsberg, in Berlin. His parents were of Jewish origin, but converted to Protestantism before he was born. An acquaintance of the family aroused an interest in mathematics in the six-year-old (“I was able to understand the proof of a sentence as a six-year-old”). He was also interested in music, played the piano and composed. After attending preschool, he was sent to boarding school outside of Berlin because of his poor health. In the rather rural Charlottenburg , he attended the reformed pedagogical Cauersche Anstalt from 1833 to 1837 (from 1834 pedagogy), a teaching and educational institution based on the principles of Fichte and Pestalozzi . From 1837 to 1842 he was at the Friedrich-Wilhelm-Gymnasium and Friedrich-Werder-Gymnasium . From 1840 he attended lectures by the mathematician Dirichlet at the University of Berlin. At the grammar school he was supported by the teacher Karl Heinrich Schellbach , he read the works of Euler , Lagrange and Gauss . In 1840 the father moved to England, but could not gain a foothold anywhere. Eisenstein and his mother followed him in the summer of 1842. They traveled through England, Wales and Ireland. His piano playing aroused admiration in Liverpool. During this time he also studied Gauss' main work on number theory, the Disquisitiones . In Dublin he met the famous mathematician and physicist William Rowan Hamilton , who gave him a paper on Abel's equation theory for publication in Berlin. In mid-June 1843 he was back in Berlin, where his parents were separated from now on. From 1843 Eisenstein moved a total of sixteen times in Berlin; from 1846 he lived separated from his mother. In 1843 his mother applied for support for him. He passed the external Abitur, whereby in his compulsory curriculum vitae he drew attention to his "hypochondriac mood" and also referred to recommendations from Dirichlet, Hamilton, Jacobi and the astronomer and secretary of the Berlin academy Johann Franz Encke . In October he enrolled at the Berlin University.


January 1844 he submitted his own work on cubic forms in two variables to Leopold Crelle for publication, in whose journal most of his work appeared. Through him he met Alexander von Humboldt in March 1844 , who promoted him, entered into correspondence with him and made numerous entries for him. From then on he received numerous donations from the king and the Ministry of Culture and the Academy: a total of 5,300 thalers, on average 250 thalers per year, whereby after his death he left 500 thalers to his parents. However, the entries had to be renewed every two to three years. At the invitation of Gauss, who praised the work sent to him, he was in Göttingen in June. There he made friends with the mathematician Moritz Stern . At the same time, in Volumes 27 and 28 of Crelles Journal in 1844, 25 of his works (23 to be more precise, and 2 problems) appeared, which made him known at once. They concerned the cubic and biquadratic reciprocity law (Gauss published only the quadratic reciprocity law in the Disquisitiones ), works on cubic forms, circular division, elliptic and Abelian functions. First honors followed: in 1845, as a third-semester student, he received an honorary doctorate from the University of Breslau at the suggestion of Kummer and Gauß suggested him for the Pour le Mérite order . He made the acquaintance of Leopold Kronecker , with whom he became friends. However, when he left university to start a career as a businessman, he was very isolated. His hypochondriac mood deteriorated and remained so until his death. At the beginning of 1846 there was a priority dispute with Jacobi: he had not mentioned him in his work on the division of the district. Jacobi called him in a letter to Bessel a "liar and (literary) thief", which also had an impact on his sponsors (Encke) in Berlin. Gauss, on the other hand, who was otherwise sparse in praise, wrote to Humboldt on April 14, 1846 that Eisenstein's talent was of the kind that "is only given to a few in every century". The 80-year-old Humboldt tried his best to counter Jacobi's influence, but was already looking for another place of activity and wrote to the Bavarian Crown Prince Maximilian and to Heidelberg.

Lectureship and end

Eisenstein completed his habilitation in 1847 and lectured at Berlin University as a private lecturer. In the summer of 1847 Bernhard Riemann heard from Eisenstein about elliptical functions. Eisenstein wrote about Riemann: "When he was here I literally followed him, but he seemed to avoid me" and attributed this to his own shyness and inaccessibility. Incidentally, he had six (elliptical functions) or two listeners (function theory), and similar in the following semesters. Eisenstein published an anthology of his work, for which Gauss wrote the foreword.

In the revolutionary year of 1848, he attended democratic clubs, but did not get involved in politics. He had been studying medicine with Johannes Müller since 1847. In letters he complained about his isolation: Dirichlet was very friendly to him, but he only felt cold politeness. On March 19, he was arrested at his house on the corner of Friedrichstrasse and Krausenstrasse during the barricade fighting with the other residents, as there had been shooting from the house. They were mistreated and taken to the citadel in Spandau, but were released the next day. During this time he gave lectures on analysis and mechanics at the university.

In March 1849 Dirichlet gave a very good judgment on him for the Ministry of Culture. The ministry also asked about circulating rumors about his "moral behavior". Apparently, however, rumors were believed that he had taken part in the revolution and cut his salary, which Humboldt was able to partially absorb, who feared a fate similar to Abel's for him . The relationship with Crelle clouded over, as he made changes to his publications without asking, with which he wanted to improve Eisenstein's style, but which in Eisenstein's opinion partially change the meaning.

In August 1850, Dirichlet and Jacobi applied for a professorship for Eisenstein, but this was rejected. He was officially informed of doubts about his teaching qualifications. He was often sick and held his lectures for the few listeners from the bed of his room. He published his irreducibility criterion and papers on higher reciprocity laws. In January 1851 he was elected a member of the Berlin Academy of Sciences after he had been proposed by Dirichlet, Jacobi and Encke. However, he was not accepted because two other candidates took the vacant positions. However, in 1851 he was, with grief, appointed a corresponding member of the Göttingen Academy of Sciences at the suggestion of Gauss . March 1852 he became a member of the Berlin Academy.

After falling down again and again due to illness in the previous years, he suffered a hemorrhage (tuberculosis) at the end of July 1852. At Humboldt's instigation, he received 500 thalers for a one-year stay in Sicily, but was too weak for that. He died in October at the age of 29 and was buried in the cemetery at Blücherplatz . The 83-year-old Humboldt gave him last escort . The grave no longer exists. Humboldt ensured the financial support of the parents, who made Humboldt's letters to their son available in 1869 in order to donate to the Humboldt Memorial from the sales proceeds.


In algebra, the Eisenstein criterion comes from him (although this was already proven by Theodor Schönemann ). Named after him are z. B. the Eisenstein numbers , the Eisenstein series and the Eisenstein functions. Access to elliptical functions via rows of iron stones was later expanded by Weierstrass and Kronecker. Eisenstein also offers new perspectives on theta functions.

In number theory he proved the cubic and biquadratic reciprocity law (from the theory of the lemniscate), gave a geometric proof of the quadratic reciprocity law (Crelle Journal Vol. 28, 1844, p. 246) and won the admiration of Gauss for it. All these laws are about the characterization of the solutions of the corresponding equations (3rd and 4th degree) in "(mod p) arithmetic". In his work on the laws of reciprocity he competes with other works by Kummer, whose ideal theory he also uses. He also tries to extend Gauss' work on square forms in the Disquisitiones to cubic forms.

Leopold Kronecker carried on many of his friend's ideas. André Weil shows in his book that Eisenstein's theories are still very topical today.

Writings and editions of works


  • Allan Adler: Eisenstein and the Jacobean Variety of Fermat curves. In: Rocky Mountain Journal of Mathematics. Volume 27, 1997, pp. 1-60.
  • Kurt-R. Biermann:  Eisenstein, Ferdinand Gotthold Max. In: New German Biography (NDB). Volume 4, Duncker & Humblot, Berlin 1959, ISBN 3-428-00185-0 , p. 420 f. ( Digitized version ).
  • Moritz CantorEisenstein, Ferdinand Gotthold Max . In: Allgemeine Deutsche Biographie (ADB). Volume 5, Duncker & Humblot, Leipzig 1877, p. 774 f.
  • Collison: The origin of the cubic and biquadratic reciprocity laws. In: Archive history of exact sciencesö Volume 16, 1977, p. 63.
  • Edwards: Kummer, Eisenstein and higher reciprocity laws. In: Koblitz (Ed.): Number theory related to Fermats last theorem. Birkhäuser, 1982.
  • Lemmermeyer: Reciprocity laws - from Euler to Eisenstein. Springer, 2000 (on Eisenstein especially p. 270 ff, with assessments by Kummer and others).
  • Ferdinand Rudio (ed.): An autobiography by Gotthold Eisenstein. In: Journal of Mathematics and Physics. Volume 40, 1895, pp. 143-168.
  • A. Hurwitz and F. Rudio (eds.): Letters from G. Eisenstein to M. A. Stern. In: Journal of Mathematics and Physics. Volume 40, 1895, pp. 169-204.
  • Norbert Schappacher : Eisenstein. In: Begehr, Koch, Kramer, Schappacher, Thiele (ed.): Mathematics in Berlin. Birkhäuser, 1998.
  • Schwermer: About reciprocity laws in number theory. In: Knörrer (Ed.): Mathematical miniatures. Volume 3, 1986.
  • Stillwell: Eisenstein's footnote. In: Mathematical Intelligencer. 1995, No. 2 (solution of equation 5th degree).
  • Peter Ullrich: About the copy of Gauss Disquisitiones from Eisenstein's possession. In: Communications of the Mathematical Society Hamburg. Volume 21, 2002, p. 35 (the copy is now in the Giessen University Library).
  • André Weil : Elliptic functions according to Kronecker and Eisenstein. Springer Verlag, Results of Mathematics and its Frontier Areas, Volume 88, 1976.
  • André Weil: On Eisenstein's copy of Gauss Disquisitiones. In: Coates (Ed.): Algebraic number theory in honor of Iwasawa. 1989. (Because suspects that Riemann got some ideas for his zeta function work from Eisenstein.)
  • André Weil: Discussion of the collected works. Bulletin AMS, Vol. 82, 1976, p. 658.

Web links

Much of Eisenstein's work, particularly Crelles Journal , where most of his work appeared, is available online.

Footnotes and sources

  1. Quoted from Kurt Biermann: Eisenstein. Crelle J. 1964
  2. A maximum of around 18, but Dirichlet, Jacobi and Steiner also gave lectures. The less gifted went to Martin Ohm .
  3. They got to the astronomer Schuhmacher in Altona, who told Gauss about it in a letter.
  4. Eisenstein: On the irreductibility and some other properties of the equation on which the division of the whole lemniscate depends. Journal for pure and applied mathematics, Volume 39, 1850, pp. 160-179.