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Loose Ends

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The Mathematics of Frobenius in Context

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Abstract

As its title suggests, this chapter is devoted to tying up several historical loose ends related to the work of Frobenius featured in the previous chapters. The first section focuses on work done by Frobenius in response to the discovery of a gap in Weierstrass’ theory of elementary divisors as it applied to families of quadratic forms. Frobenius gave two solutions to the problem of filling the gap. The first drew upon the results and analogical reasoning used in his arithmetic theory of bilinear forms and its application to elementary divisor theory (Chapter 8).

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Notes

  1. 1.

    This fact and the other information preliminary to Frobenius’ discovery of a proof is drawn from Frobenius’ account in 1894 [203, pp. 577–578].

  2. 2.

    In the case of algebraic integers, some results require excluding p a prime divisor of 2 [203, p. 584] in the definition of “regular with respect to p.”

  3. 3.

    On Kronecker’s work with fields, see Purkert’s study [491].

  4. 4.

    Frobenius’ theorem actually specifies that \(\deg \chi = m - 1\), where m is the degree of the minimal polynomial of U. This additional information can be inferred from Frobenius’ proof; see below.

  5. 5.

    This is indeed correct, as Frobenius showed in his 1878 paper [181, p. 355], assuming that when \(f(t) = p(t)/q(t)\), detq(A) ≠ 0, so that [q(A)] − 1 exists.

  6. 6.

    If L 2 = M, then \({L}^{4} = {M}^{2} = 0\). This means that the characteristic roots of L must all be 0 and so (since L is 2 ×2) \(\varphi (t) = {t}^{2}\) is the characteristic polynomial of L. The Cayley–Hamilton theorem then implies \({L}^{2} = 0\not =M\), and so \(\sqrt{M}\) does not exist.

  7. 7.

    For the personal and institutional background to Sylvester’s brief flurry of interest in matrix algebra, see [462, pp. 135–138]. A fairly detailed mathematical discussion of Sylvester’s work on matrix algebra is given in [270, §6].

  8. 8.

    I refer to Lagrange’s attempt to extend his elegant generic solution to \(\ddot{y} + Ay = 0\), y(0) = y 0, A n ×n, to the case in which \(f(\rho ) =\det {(\rho }^{2} + A)\) has one root of multiplicity two. See Section 4.2.1.

  9. 9.

    See in this connection Frobenius’ use of power and Laurent series in his proof of his minimal polynomial theorem (Theorem 7.2) and in his proof that a real orthogonal matrix can be diagonalized (Theorem 7.15).

  10. 10.

    Frobenius’ proof [208, pp. 697ff.] is expressed somewhat more generally than expounded here so as to allow a brief discussion of the problematic case det U = 0 as well as other functions of a matrix.

  11. 11.

    Frobenius proved that if S is a matrix with the property that the sequence S j, j = 0, 1, 2, …, has only a finite number of distinct terms, then all characteristic roots of S are either 0 or roots of unity, and the elementary divisors corresponding to the roots of unity are all linear [181, Satz VI, p. 357].

  12. 12.

    See in this connection [270, p. 107n.15].

  13. 13.

    Regarding Muth’s life and work, see [463].

  14. 14.

    In the preface, Muth wrote that he had been encouraged by the fact that “From the outset my undertaking was of special interest to several outstanding experts in the theory of elementary divisors,” namely Frobenius, S. Gundelfinger, and K. Hensel (Kronecker’s former student) [450, p. iv].

  15. 15.

    See, e.g., [25, pp. 297–301], [127, pp. 120–125], [567, pp. 130–131], [430, pp. 60–61], [240, v. 2, 41–42].

  16. 16.

    Kronecker ignored the work of Smith, even though Frobenius referenced it throughout his paper.

  17. 17.

    For a contemporary rendition, see, e.g., [141, pp. 459ff.].

  18. 18.

    Weber’s Lehrbuch der Algebra in its various editions from 1895 onward did not treat elementary divisor theory.

  19. 19.

    Bôcher also mentioned (1) λ-polynomials with integer coefficients and (2) polynomials in several variables; but it is unclear what he thought could be proved in these two cases, neither of which involves a principal ideal domain.

  20. 20.

    Let \(A ={\biggl ( \begin{array}{ccc} \rho _{1}\rho _{2}\! & \!0 \\ 0\! & \!1 \\ \end{array} \biggr )}\) and \(B ={\biggl ( \begin{array}{ccc} \rho _{1}\! & \!0 \\ 0\! & \!\rho _{2} \\ \end{array} \biggr )}\). Then A and B have the same invariant factors but B = P A Q is impossible, because it implies detPdetQ = 1 as well as that detP depends on the ρ i and vanishes for \(\rho _{1} =\rho _{2} = 0\) [425, p. 107].

  21. 21.

    On the rationality group and Picard–Vessiot theory, see Gray’s historical account [255, pp. 267ff.].

  22. 22.

    Loewy refers to Schlesinger’s 1908 book on differential equations [514, pp. 157–158], which may have suggested to him the value of this connection for his own work, as described below.

  23. 23.

    The main results of Loewy’s theory are outlined in [427]. His most complete exposition is contained in a paper published in Mathematische Zeitschrift in 1920 and dedicated to Ludwig Stickelberger on the occasion of his 70th birthday [429]. The 1917 paper, however, brings out more fully the connections with Frobenius’ theory.

  24. 24.

    This result is implicit in [427] and explicit in [429, p. 102].

  25. 25.

    Krull’s companion matrices were defined as the negatives of Loewy’s because Krull defined the characteristic polynomial as det(t I − A), rather than as det(t I + A), as with Loewy.

  26. 26.

    These papers are [4]–[7] in Krull, Abhandlungen 1. For an overview of their contents, as well as Krull’s subsequent work on the theory of ideals, see P. Ribenboim’s essay on Krull [494, pp. 3ff.].

  27. 27.

    Despite his considerable mathematical talent, Remak, a Jew, did not have a correspondingly successful career as a mathematician and was eventually deported to Auschwitz, where he perished. For more on Remak’s life and work, see [440].

  28. 28.

    In the terminology and notation of Remak, \(\mathfrak{H}\) is the direct product \(\mathfrak{H} = \mathfrak{A} \times \mathfrak{B}\) of nontrivial subgroups \(\mathfrak{A}\), \(\mathfrak{B}\) if (1) every \(g \in \mathfrak{G}\) is expressible as g = a b, with \(a \in \mathfrak{A}\) and \(b \in \mathfrak{B}\); (2) \(\mathfrak{A} \cap \mathfrak{B} =\{ E\}\); (3) every \(a \in \mathfrak{A}\) commutes with every \(b \in \mathfrak{B}\). When \(\mathfrak{H}\) is abelian, this agrees with the definition of \(\mathfrak{H} = \mathfrak{A} \times \mathfrak{B}\) given by Frobenius and Stickelberger in 1878. Condition (3) must be added for nonabelian groups.

  29. 29.

    An automorphism σ of \(\mathfrak{H}\) is central if h  − 1 σ(h) is in the center of \(\mathfrak{H}\) for all \(h \in \mathfrak{H}\) [493, p. 293], a notion still in use today.

  30. 30.

    The ascending chain condition for ideals had been introduced by Noether in her above-mentioned fundamental paper of 1921 [455, Satz I, p. 30] and goes back to Dedekind, as she noted. All Krull’s early papers on ideals cite this paper as a basic reference. Nine months before submitting his paper [375] on generalized abelian groups, Krull had submitted a paper with a descending chain condition on the successive powers of an ideal [374, p. 179, (f)]. As for Noether, her important work on what are now called Dedekind rings [456, 457], involved a descending chain condition. A bit later, Artin used a descending chain condition in his study of what are now called Artinian rings.

  31. 31.

    Krull spoke simply of indecomposable subgroups, but he meant the analogue of what Remak had called “directly indecomposable” subgroups.

  32. 32.

    See [375, p. 191 n.34]. Krull was referring to Loewy’s results on matrix complexes (described above) as they apply to the completely reducible operators of Loewy’s Theorem 16.8. Of course, it should be kept in mind that although Krull’s theorem applies to any \(\mathfrak{A}\), the \(\mathfrak{D}_{ii}\) are simply directly indecomposable and not necessarily irreducible in Loewy’s sense.

  33. 33.

    Krull did not use the terminology of linear transformations and vector spaces, which was soon brought into linear algebra primarily through the influence of the work of Weyl (see below).

  34. 34.

    Krull’s matrix of Θ with respect to a basis is the transpose of the usual definition. Thus he gets the transpose of C i  [376, p. 26], which agrees with how companion matrices are defined in his doctoral dissertation [372, p. 58].

  35. 35.

    Lasker reserved the term “ideal” for an ideal in Z[x 1, …, x n ].

  36. 36.

    Châtelet also extended his notion of module to “smireal” n-dimensional spaces [88, pp. 10, 25].

  37. 37.

    In §§106–107, van der Waerden also allowed \(\mathfrak{R}\) to be what one might call a “noncommutative Euclidean domain” [568, p. 120].

  38. 38.

    That is, detU, which belongs to \(\mathfrak{R}\), has an inverse in \(\mathfrak{R}\). Thus \({U}^{-1} = {(\det U)}^{-1}\mathrm{Adj}\,U\) is a matrix over \(\mathfrak{R}\), and \(x = u{U}^{-1}\).

  39. 39.

    Sperner explained in the preface that the subject matter of the book was as Schreier intended, although the ordering of the material and some proofs were changed in part to achieve greater simplicity.

  40. 40.

    In Space–Time–Matter, first published in 1921, Weyl developed the machinery of tensor algebra within the context of abstract vector spaces. In his 1923 monograph Mathematical Analysis of the Space Problem [601, Anhang 12], Weyl developed elementary divisor theory over \(\mathbb{C}\) within the same context and so made the concept of a linear transformation acting on a vector space fundamental. For more on Weyl and the space problem, see [518, §2.8] and [276, §11.2].

  41. 41.

    For a discussion of Schreier’s important work on continuous groups, see [276, pp. 497ff.].

  42. 42.

    Under Schmidt’s influence, van der Waerden devoted an entire chapter [568, Ch. 6] to groups with operators [569, p. 34].

  43. 43.

    As with Theorem  16.11, \(\mathfrak{R}\) can also be a “noncommutative Euclidean domain.” See the citation at that theorem.

References

  1. M. Bôcher. Introduction to Higher Algebra. Macmillan, New York, 1907. Republished by Dover Publications, New York, 1964. German translation as [26].

    Google Scholar 

  2. M. Bôcher. Einführung in die höhere Algebra. Teubner, Leipzig, 1910. Translated by H. Beck.

    Google Scholar 

  3. A. Cayley. A memoir on the theory of matrices. Phil. Trans. R. Soc. London, 148:17–37, 1858. Reprinted in Papers 2, 475–496.

    Google Scholar 

  4. A. Châtelet. Leçons sur la théorie des nombres. (Modules. Entiers algébrique. Réduction continuelle.). Gauthier–Villars, Paris, 1913.

    Google Scholar 

  5. L. E. Dickson. Modern Algebraic Theories. Sandborn, Chicago, 1926.

    MATH  Google Scholar 

  6. D. Dummit and R. Foote. Abstract Algebra. Wiley, 2nd edition, 1999.

    Google Scholar 

  7. G. Frobenius. Über den Begriff der Irreductibilität in der Theorie der linearen Differentialgleichungen. Jl. für die reine u. angew. Math., 76:236–270, 1873. Reprinted in Abhandlungen 1, 106–140.

    Google Scholar 

  8. G. Frobenius. Über lineare Substitutionen und bilineare Formen. Jl. für die reine u. angew. Math., 84:1–63, 1878. Reprinted in Abhandlungen 1, 343–405.

    Google Scholar 

  9. G. Frobenius. Theorie der linearen Formen mit ganzen Coefficienten. Jl. für die reine u. angew. Math., 86:146–208, 1879. Reprinted in Abhandlungen 1, 482–544.

    Google Scholar 

  10. G. Frobenius. Theorie der linearen Formen mit ganzen Coefficienten (Forts.). Jl. für die reine u. angew. Math., 88:96–116, 1880. Reprinted in Abhandlungen 1, 591–611.

    Google Scholar 

  11. G. Frobenius. Über die Elementartheiler der Determinanten. Sitzungsberichte der Akademie der Wiss. zu Berlin, pages 7–20, 1894. Reprinted in Abhandlungen 2, 577–590.

    Google Scholar 

  12. G. Frobenius. Über die cogredienten Transformationen der bilinearen Formen. Sitzungsberichte der Akademie der Wiss. zu Berlin, pages 7–16, 1896. Reprinted in Abhandlungen 2, 695–704.

    Google Scholar 

  13. G. Frobenius. Über vertauschbare Matrizen. Sitzungsberichte der Akademie der Wiss. zu Berlin, pages 601–614, 1896. Reprinted in Abhandlungen 2, 705–718.

    Google Scholar 

  14. G. Frobenius and I. Schur. Über die äquivalenz der Gruppen linearer Substitutionen. Sitzungsberichte der Akademie der Wiss. zu Berlin, pages 209–217, 1906. Reprinted in Frobenius, Abhandlungen 3, 378–386.

    Google Scholar 

  15. F. Gantmacher. Matrix Theory. AMS Chelsea Publishing, 2000. This work, published in two volumes, is an English translation of Gantmacher’s Teoriya Matrits (Moscow, 1953). It first appeared in 1959.

    Google Scholar 

  16. J. Gray. Linear Differential Equations and Group Theory from Riemann to Poincaré. Birkhäuser, Boston, 2nd edition, 2000.

    MATH  Google Scholar 

  17. O. Haupt. Einführung in die Algebra, volume 2. Akademische Verlaggesellschaft, Leipzig, 1929. The theory of generalized abelian groups and its application to elementary divisor theory is presented by Krull in an appendix (pp. 617–629).

    Google Scholar 

  18. T. Hawkins. Another look at Cayley and the theory of matrices. Archives internationales d’histoire des sciences, 26:82–112, 1977.

    MathSciNet  Google Scholar 

  19. T. Hawkins. Emergence of the Theory of Lie Groups. An Essay on the History of Mathematics 1869–1926. Springer, New York, 2000.

    Google Scholar 

  20. L. Heffter. Einleitung in die Theorie der linearen Differentialgleichungen mit einer unabhängigen Variable. B. G. Teubner, Leipzig, 1894.

    Google Scholar 

  21. K. Hensel. Ueber die Elementheiler componirter Systeme. Jl. für die reine u. angew. Math., 114:109–115, 1895.

    Google Scholar 

  22. K. Hentzelt and E. Noether. Bearbeitung von K. Hentzelt: zur Theorie der Polynomideale und Resultanten. Math. Ann., 88:53–79, 1923. Reprinted in E. Noether, Abhandlungen, 409–435.

    Google Scholar 

  23. C. Jordan. Mémoire sur les équations différentielles linéaires à intégrale algébrique. Jl. für die reine u. angew. Math., 84:89–215, 1878. Reprinted in Oeuvres 2, 13–139.

    Google Scholar 

  24. A. Krazer. Lehrbuch der Thetafunktionen. Teubner, Leipzig, 1903. Reprinted by Chelsea Publishing Company (New York, 1970).

    Google Scholar 

  25. L. Kronecker. Über die congruenten Transformation der bilinearen Formen. Monatsberichte der Akademie der Wiss. zu Berlin, pages 397–447, 1874. Presented April 23, 1874. Reprinted in Werke 1, 423–483.

    Google Scholar 

  26. L. Kronecker. Über die Composition der Systeme von n 2 Grössen mit sich selbst. Sitzungsberichte der Akademie der Wiss. zu Berlin, pages 1081–1088, 1890. Reprinted in Werke 31, 463–473.

    Google Scholar 

  27. L. Kronecker. Algebraische Reduction der Schaaren bilinearer Formen. Sitzungsberichte der Akademie der Wiss. zu Berlin, pages 1225–1237, 1890. Reprinted in Werke 32, 141–155.

    Google Scholar 

  28. L. Kronecker. Algebraische Reduction der Schaaren quadratischer Formen. Sitzungsberichte der Akademie der Wiss. zu Berlin, pages 1375–1388, 1890. Reprinted in Werke 32, 159–174.

    Google Scholar 

  29. L. Kronecker. Algebraische Reduction der Schaaren quadratischer Formen. Sitzungsberichte der Akademie der Wiss. zu Berlin, pages 9–17, 34–44, 1891. Reprinted in Werke 32, 175–198.

    Google Scholar 

  30. L. Kronecker. Reduction der Systeme von n 2 ganzahlige Elementen. Jl. für die reine u. angew. Math., 107:135–136, 1891. Reprinted in Werke 4, 123–124.

    Google Scholar 

  31. W. Krull. Über Begleitmatrizen und Elementartheiler. Inauguraldissertation Universität Freiburg i. Br., 1921. First published in Krull’s Abhandlungen 1 (1999), 55–95.

    Google Scholar 

  32. W. Krull. Algebraische Theorie der Ringe. I. Math. Ann., 88:80–122, 1922. Reprinted in Abhandlungen 1, 80–122.

    Google Scholar 

  33. W. Krull. Algebraische Theorie der Ringe. II. Math. Ann., 91:1–46, 1924. Reprinted in Abhandlungen 1, 166–211.

    Google Scholar 

  34. W. Krull. Über verallgemeinerte endliche Abelsche Gruppen. Mathematische Zeitschrift, 23:161–196, 1925. Reprinted in Papers 1, 263–298.

    Google Scholar 

  35. W. Krull. Theorie und Anwendung der verallgemeinten Abelschen Gruppen. S’ber. Akad. Wiss. Heidelberg, Math.-Natur. Kl., 1:1–10, 1926. Reprinted in Papers 1, 299–328.

    Google Scholar 

  36. E. Landau. Ein Satz über die Zerlegung homogener linearer Differentialausdrücke in irreducible Factoren. Jl. für die reine u. angew. Math., 124, 1902.

    Google Scholar 

  37. R. Lipschitz. Beweis eines Satzes aus der Theorie der Substitutionen. Acta Mathematica, 10:137–144, 1887.

    Article  MathSciNet  MATH  Google Scholar 

  38. A. Loewy. Ueber die irreduciblen Factoren eines linearen homogenen Differentialausdruckes. Berichte über d. Verh. d. Sächsischen Gesell. der Wiss., math. -phys. Klasse, pages 1–13, 1902.

    Google Scholar 

  39. A. Loewy. Über die Reducibilität der Gruppen linearer homogener Substitutionen. Trans. American Math. Soc., 4:44–64, 1903.

    MathSciNet  MATH  Google Scholar 

  40. A. Loewy. Über reduzible lineare homogene Differentialgleichungen. Math. Ann., 56:549–584, 1903.

    Article  MathSciNet  MATH  Google Scholar 

  41. A. Loewy. Kombinatorik, Determinanten und Matrices. In P. Epstein and H. E. Timmerding, editors, Repertorium der höheren Mathematik, volume 1, chapter 2. Leipzig and Berlin, 1910.

  42. A. Loewy. Über lineare homogene Differentialsysteme und ihre Sequenten. Sitzungsberichte der Heidelberger Akademie der Wissenschaften, math.–naturwis. Kl., Abt. A, Abhandlung 17, 1913.

    Google Scholar 

  43. A. Loewy. Die Begleitmatrix eines linearen homogenen Differentialausdruckes. Nachrichten von der Königlichen Gesellschaft der Wissenschaften und der Georg-Augustus-Universität zu Göttingen, pages 255–263, 1917.

    Google Scholar 

  44. A. Loewy. Über Matrizen- und Differentialkomplexe. Math. Ann., 78:1–51, 1917.

    Article  MathSciNet  MATH  Google Scholar 

  45. A. Loewy. Begleitmatrizen und lineare homogene Differentialausdrücke. Mathematische Zeitschrift, 7:58–125, 1920.

    Article  MathSciNet  MATH  Google Scholar 

  46. C. C. MacDuffee. The Theory of Matrices. Springer, Berlin, 1933.

    Book  Google Scholar 

  47. U. Merzbach. Robert Remak and the estimation of units and regulators. In S. S. Demidov et al, editor, Amphora: Festschrift für Hans Wußing zu seinem 65. Geburtstag, pages 481–552. Birkhäuser, Berlin, 1992.

    Google Scholar 

  48. P. Muth. Theorie und Anwendung der Elementartheiler. Teubner, Leipzig, 1899.

    Google Scholar 

  49. E. Noether. Idealtheorie in Ringbereichen. Math. Ann., 83:24–66, 1921. Reprinted in Abhandlungen, 354–366.

    Google Scholar 

  50. E. Noether. Abstrakter Aufbau der Idealtheorie im algebraischen Zahlkörper. Jahresbericht der Deutschen Mathematiker-Vereinigung, 33:102, 1924. Reprinted in Abhandlungen, p. 102.

    Google Scholar 

  51. E. Noether. Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern. Math. Ann., 96:26–61, 1926–27. Reprinted in Abhandlungen, 493–528.

    Google Scholar 

  52. E. Noether. Hypercomplexe Grössen und Darstellungstheorie. Mathematische Zeitschrift, 30:641–692, 1929. Reprinted in Abhandlungen, 563–614.

    Google Scholar 

  53. E. Noether and W. Schmeidler. Moduln in nichtkommutativen Bereichen, insbesondere aus Diffzenzausdrücken. Mathematische Zeitschrift, 8:1–35, 1920. Reprinted in Abhandlungen, 318–352.

    Google Scholar 

  54. K. H. Parshall and D. Rowe. The Emergence of the American Mathematical Research Community, 1876–1900: J. J. Sylvester, Felix Klein, and E. H. Moore. History of Mathematics, Vol. 8. American Mathematical Society, 1994.

    Google Scholar 

  55. M. Pasch. Peter Muth. Jahresbericht der Deutschen Mathematiker-Vereinigung, 18:454–456, 1909.

    Google Scholar 

  56. W. Purkert. Zur Genesis des abstrakten Körperbegriffs. 1. Teil. NTM-Schriftenreihe für Geschichte der Naturwiss., Technik, und Med., 8:23–37, 1971.

    Google Scholar 

  57. R. Remak. Über die Zerlegung der endlichen Gruppen in direkte unzerlegbare Faktoren. Jl. für die reine u. angew. Math., 139:293–308, 1911.

    MATH  Google Scholar 

  58. P. Ribenboim. Wolfgang Krull—Life, Work and Influence. In Wolfgang Krull Gesammelte Abhandlingen, volume 1, pages 1–20. Walter de Gruyter, Berlin, 1999.

    Google Scholar 

  59. L. Schlesinger. Handbuch der Theorie der linearen Differentialgleichungen, volume 1. B. G. Teubner, 1895.

    MATH  Google Scholar 

  60. L. Schlesinger. Vorlesungen über Differentialgleichungen. B. G. Teubner, Leipzig, 1908.

    Google Scholar 

  61. O. Schmidt. Über unendliche Gruppen mit endlicher Kette. Jl. für die reine u. angew. Math., 29:34–41, 1928–1929.

    Google Scholar 

  62. E. Scholz. Historical aspects of Weyl’s Raum–Zeit–Materie. In E. Scholz, editor, Hermann Weyl’s Raum–Zeit–Materie and a General Introduction to His Scientific Work. Birkhäuser, 2000.

    Google Scholar 

  63. O. Schreier and E. Sperner. Vorlesungen über Matrizen. B. G. Teubner, Leipzig, 1932.

    Google Scholar 

  64. L. Stickelberger. Ueber Schaaren von bilinearen und quadratischen Formen. Jl. für die reine u. angew. Math., 86:20–43, 1879.

    Google Scholar 

  65. J. J. Sylvester. On the equation to the secular inequalities in the planetary theory. Phil. Mag., 16:110–11, 1883. Reprinted in Papers, v. 4, 110–111.

    Google Scholar 

  66. H. W. Turnbull and A. C. Aitken. An Introduction to the Theory of Canonical Matrices. Blackie and Son, London & Glasgow, 1932.

    Google Scholar 

  67. B. L. van der Waerden. Moderne Algebra, volume 2. Springer, Berlin, 1931.

    Google Scholar 

  68. B. L. van der Waerden. On the sources of my book Moderne Algebra. Historia Mathematica, 2:31–40, 1975.

    Google Scholar 

  69. J. H. M. Wedderburn. Lectures on Matrices. American Mathematical Society, New York, 1934.

    Google Scholar 

  70. K. Weierstrass. Zur Theorie des quadratischen und bilinearen Formen. Monatsberichte der Akademie der Wiss. zu Berlin, pages 311–338, 1868. Reprinted with modifications in Werke 2, 19–44.

    Google Scholar 

  71. H. Weyl. Mathematische Analyse des Raumproblems. Springer, Berlin, 1923.

    Book  Google Scholar 

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Hawkins, T. (2013). Loose Ends. In: The Mathematics of Frobenius in Context. Sources and Studies in the History of Mathematics and Physical Sciences. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6333-7_16

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