Abstract
The correspondence between the two Scandinavian mathematicians Julius Petersen and Ludvig Sylow is interesting, not only because it sheds light on the researches of the two correspondents, but also because it gives a good impression of the general state of affairs in algebra, more specifically group theory and Galois theory during the early and middle 1870’s, as seen from the nothern fringe of the European mathematical scene. It thus complements Hans Wussing’s vivid and detailed account of this period in his classical The Genesis of the Abstract Group Concept [Wussing 1969/84]. Wussing shows that around 1870 the mathematical avantgarde had emancipated group theory as a separate discipline independent of its use in the theory of equations, thereby paving the way for the introduction of the abstract group concept around 1880. The correspondence between Petersen and Sylow, however, reminds us of the fact that it takes time for such conceptual innovations to filter down through the system. Indeed, for Petersen and Sylow, as well as many of their colleagues, groups were still tied to equations, if they were known at all, and the abstract definition was a far cry of the future. Waterhouse [1980], and in particular Schar-Lau [1988], have emphasized that Sylow’s proof of the famous theorems named after him, built crucially on the fact that he considered groups as Galois groups of a certain equation. In this connection, Scharlau quotes the laconic proof in Petersen’s algebra book [Petersen 1877] of the theorem stating that every group can be considered as a Galois group.
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References
Abel, N. H.: Recherches sur les fonctions elliptiques. Journal für die reine und angewandte Mathematik 2 (1827), 101–181; 3 (1828), 160–190; Oeuvres (ed. Sylow and Lie) I, 263–388.
Abel, N. H.: Mémoire sur une classe particulière d’équations résolubles algébriquement. Journal für die reine und angewandte Mathematik 4 (1829), 131–156; Oeuvres (ed. Sylow and Lie) I, 478–507.
Abel, N. H.: Sur la résolution algébrique des équations. In: Oeuvres Complètes de N. H. Abel, ed. B. Holmboe, Christiania, 1839, vol. II, 185–209; Oeuvres (ed. Sylow and Lie) II, 217–243.
Christiansen, M., LÜTZEN, J., Sabidussi, G., and Toft, B.: Julius Petersen. Bibliography. To appear in: Discrete Mathematics, 100.
Galois, E.: Analyse d’un mémoire sur la résolution algébrique des équations. Bulletin des Sciences mathématiques 13 (1830), 271; Oeuvres (ed. Picard) 11–12.
Galois, E.: Mémoire sur les conditions de résolubilité des équations par radicaux. Journal de Mathématiques pures et appliquées 11 (1846), 417–433. Slightly revised version of a paper communicated to the Académie des Sciences in Paris in 1831; Oeuvres (ed. Picard), 33–50.
Galois, E.: Des équations primitives qui sont solubles par radicaux. Journal de Mathématiques pures et appliquées 11 (1846), 434–444; Oeuvres (ed. Picard), 51–61.
Gauss, C. F.: Disquisitiones Arithmeticae, Braunschweig 1801; Werke 1. English translation, New Haven and London 1966.
Hilbert, D.: Über die Irreduzibilität ganzer rationaler Funktionen mit ganzzahligen Koeffizienten. Journal für die reine und angewandte Mathematik 110 (1892), 104–129; Ges. Abhandlungen II, 264–286.
Jordan, C: Traité des substitutions et des équations al-gébriques. Paris, 1870.
Juel, C: En dansk Matematiker. Matematisk Tidsskrift A (1923), 85–95.
Kragemo, H. B.: Ludvig Sylow. Norsk Matematisk Tidsskrift 15 (1933), 73–99.
Kragemo, H. B.: Bibliographie der Schriften Ludvig Sylows. Norsk Matematisk Forenings Skrifter, ser. II nr. 1–12 (1933), 25–30.
Kronecker, L.: Über die algebraisch auflösbaren Gleichungen I. Monatsberichte der Königlich Preußischen Akademie der Wissenschaften (1853), 365–374; Werke 4, 1–11.
Lagrange, J. L.: Réflexions sur la résolution algébrique des équations. Nouveaux Mémoires de l’Académie royale des Sciences et Belles-Lettres de Berlin, 1770–1771. Oeuvres, vol. 3, 205–421.
LÜTZEN, J.: Joseph Liouville 1809–1882. Master of Pure and Applied Mathematics. New York (Springer Verlag), 1990.
Lützen, J., Sabidussi, G., Toft, B.: Julius Petersen 1839–1910. A Biography. To appear in: Discrete Mathematics 100.
Petersen, J.: Methoder og Theorier til Løsning af geometriske Konstruktionsopgaver. Kjøbenhavn (Karl Schønbergs Forlag), 1866.
Petersen, J.: Ligningen hvis Rødder ere nte Potens af Rødderne i en given Ligning. Tidsskrift for Mathematik (2) 3 (1867), 46.
Petersen, J.: Om Ligninger des løses ved Kvadratrod, med Anvendelse paa Problemers Løsning ved Passer og Lineal. Kjøbenhavn (C. Ferslew & Co.), 1871. (Petersen’s dissertation).
Petersen, J.: De algebraiske Ligningers Theori. Kjøben-havn (Høst & Søn), 1877. German edition 1878, Italian edition 1890–91, French edition 1897.
Petersen, J.: Methoder og Theorier til Løsning af Geometriske Konstruktionsopgaver. 2. ed. Kjøbenhavn (Karl Schønbergs Forlag), 1879.
Petersen, J.: Die Theorie der regulären Graphs. Acta Mathematica 15 (1891), 193–220.
Scharlau, W.: Die Entdeckung der Sylow-Sätze. Historia Mathematica 15 (1988), 40–52.
Serret, J. A.: Cours d’Algèbre supérieure. 3. ed. Paris, 1866.
Skolem, T.: Ludvig Sylow og hans videnskabelige arbeider. Norsk Matematisk Tidsskrift 1 (1919), 1–13.
Skolem, T.: Sylow und seine wissenschaftlichen Arbeiten. Norsk Matematisk Forenings Skrifter, Ser. II, nr. 1–12 (1933), 14–24.
Størmer, C: Gedächtnisrede auf Professor Dr. P. L. M. Sylow. Norsk Matematisk Forenings Skrifter, Ser. II, nr. 1–12 (1933), 7–13.
Sylow, L.: Om algebraisk Opløsning av Ligninger. Forhandlinger ved de skandinaviske Naturforskeres ottende Møde i Kjøbenhavn fra den 8de til den 14de Juli 1860. Kjøbenhavn 1861, 536–548.
Sylow, L.: Om Systemer af konjugerte Substitutioner, der kunne tilhøre irreduktible Ligninger, hvis Grad er Primtal. Forhandlinger i Videnskabs-Selskabet i Christiania Aar 1867. Christiania 1868, 105–122.
Sylow, L.: Théorèmes sur les groupes de substitutions. Mathematische Annalen 5 (1872), 584–594.
Wantzel, P. L.: Recherches sur les moyens de reconnaître si un problème de géométrie peut se résoudre avec la règle et le compas. Journal de Mathématiques pures et appliquées 2 (1837), 366–372.
Waterhouse, W. C: The Early Proofs of Sylow’s Theorem. Archive for History of Exact Sciences 21 (1980), 279–290.
Wussing, H.: Die Genesis des abstrakten Gruppenbegriffes. Berlin (VEB Deutscher Verlag der Wissenschaften), 1969. English translation: The Genesis of the Abstract Group Concept. Cambridge/Mass. (MIT Press), 1984. References to the English edition.
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Lützen, J. (1992). The Mathematical Correspondence between Julius Petersen and Ludvig Sylow. In: Demidov, S.S., Rowe, D., Folkerts, M., Scriba, C.J. (eds) Amphora. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8599-7_21
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DOI: https://doi.org/10.1007/978-3-0348-8599-7_21
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