Abstract
Ferdinand Georg Frobenius was born in Berlin on 26 October 1849. He was a descendant of a family stemming from Thüringen, a former state in central Germany and later a part of East Germany. Georg Ludwig Frobenius (1566–1645), a prominent Hamburg publisher of scientific works, including those written by himself on philology, mathematics, and astronomy, was one of his ancestors. His father, Christian Ferdinand, was a Lutheran pastor, and his mother, Christiane Elisabeth Friedrich, was the daughter of a master clothmaker.
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Notes
- 1.
- 2.
A listing of the semesters in which Weierstrass lectured on abelian integrals and functions is given in vol. 3 of his Mathematische Werke, the volume containing of a version of these lectures.
- 3.
According to Frobenius [202, p. 712], in 1858 Kronecker sent a (now lost) manuscript containing some such extension to Dirichlet and Kummer. In 1882, on the occasion of the fiftieth anniversary of Kummer’s doctorate, he presented a sketch of his ideas as they stood then [363], but they were difficult to follow. Edwards [150] has given a possible reconstruction of what Kronecker had in mind.
- 4.
See Frobenius’ discussion of this and related results [202, pp. 712–713].
- 5.
This is clear from the vita at the end of his doctoral dissertation [171, p. 34], as is the fact that his other courses at Berlin were in physics and philosophy.
- 6.
In 1875 Frobenius published a paper [176] that was an outgrowth of the work he had done on the prize problem.
- 7.
The occasion was a proposal to the Prussian minister of culture of a new associate professorship in mathematics, with Frobenius as the choice to fill the new position. The entire document is given by Biermann [22, pp. 189ff.]; the quotation below is from pp. 190–191.
- 8.
- 9.
See [22, p. 85], where the quotations below about Frobenius’ dissertation and doctoral examination are given in the original German.
- 10.
I am grateful to my colleague Dan Weiner for supplying the translations.
- 11.
The information in this paragraph is drawn from a document published by Biermann [22, p. 191].
- 12.
In discussing Fuchs’ work, as well as the related work of Frobenius, I have drawn upon Gray’s more definitive account [255, Chs. II–III].
- 13.
Thomé should not be confused with Carl Johannes Thomae (1840–1921), who also worked in complex function theory but had received his doctorate from Göttingen in 1864 and then spent two semesters attending Weierstrass’ lectures in Berlin before becoming an instructor in Göttingen in 1866. In 1874 he became a full professor at the University of Freiburg, where he spent the rest of his career.
- 14.
For an account of Schwarz’s work see [255, pp. 70–77].
- 15.
- 16.
Let \(g(x) \in \mathbb{K}[x]\) denote the minimal polynomial of V with \(m =\deg g(x)\). Then since \(\mathbb{K}(a_{1},\ldots,a_{n}) = \mathbb{K}[V ]\), each root a i of f(x) is uniquely expressible as a polynomial in V, \(a_{i} =\phi _{i}(V )\), where \(\phi (x) \in \mathbb{K}[x]\) has degree at most m − 1. If \(V ^{\prime},V ^{\prime\prime},\ldots,{V }^{(m-1)}\) are the other roots of the minimal polynomial g(x), then G consists of m permutations \(\sigma _{1},\ldots,\sigma _{m}\) of \(a_{1},\ldots,a_{n}\) with σ k the mapping that takes the root \(a_{i} =\phi _{i}(V )\) to \(\phi _{i}({V }^{(k)})\), which is also a root a i ′ of f(x), so \(\sigma _{k}: a_{i} \rightarrow a_{i^{\prime}}\) for i = 1, …, n and \(k = 0,\ldots,m - 1\). In the nineteenth century, permutations in the sense of mappings of a finite set of symbols were called “substitutions.” In the example at hand, σ k substituted the arrangement (or permutation) a 1′ , …, a n ′ for the original arrangement a 1, …, a n . Readers interested in a more detailed and historically accurate portrayal of Galois’ ideas, including a detailed working out of Galois’ sketchy remarks about the construction and properties of V, should consult Edwards’ lucid exposition of Galois’ memoir [148], which includes as appendix an annotated English translation of the memoir.
- 17.
Perhaps what he meant is illustrated by the example \(f(y,z) = {z}^{n}{y}^{n} - 1 = 0\). In this case, \({y}^{n} = 1/{z}^{n}\), and so letting z → z 0 = 0, y n → ∞. Of course, \(f(y, 0) = -1\) is a polynomial of degree zero with no roots. What Frobenius meant more generally was perhaps that at singular points z = z 0, a n (z 0) = 0.
- 18.
In what follows, I focus on the group G and omit the subgroups G ∗ that result by adjunction of function elements, although Theorem IX applies more generally to G ∗ .
- 19.
Assume that \(f(y,z) \in \mathbb{K}[y]\), \(\mathbb{K} = \mathbb{C}(z)\), is irreducible and that a 1, …, a m are the singular points a, meaning the points at which f(y, a) has a multiple root or has degree in y less than n, i.e., a n (a) = 0. By the identity theorem, the singular points are finite in number, say a 1, …, a m . Let Γ denote a non-self-intersecting polygonal line joining \(a_{1},\ldots,a_{m},\infty \), and set \(D = \mathbb{C} \sim \Gamma \). Then D is an open, connected, and simply connected set, and by the Weierstrass monodromy theorem each locally defined root y j (z) has an extension Y j (z) to D that is single-valued and analytic. (See, e.g., [348, pp. 126–127].) The identity theorem implies that f(z, Y j (z)) = 0 throughout D. That same theorem implies that Frobenius’ Theorem IX holds with the y j (z) replaced by the Y j (z). Thus if \(\mathbb{L}\) is the field of all meromorphic functions defined on D and expressible rationally in terms of Y 1, …, Y n , then \(\mathbb{L} \supset \mathbb{K} = \mathbb{C}(z)\) and \(\mathbb{L} = \mathbb{K}(Y _{1},\ldots,Y _{n})\) is a splitting field for f(y, z) over \(\mathbb{K}\). Furthermore, by Frobenius’ Theorem IX (as extended to Y 1, …, Y n ), his group G can be identified with \(\mathrm{Aut}(\mathbb{L}, \mathbb{K})\). Thus G is a bona fide Galois group.
- 20.
The French word is substitution, which was used in the nineteenth century for permutations in the modern sense of mappings, as indicated in the earlier footnote on Galois’ definition of the group associated to a polynomial equation.
- 21.
- 22.
Suppose f is irreducible over \(\mathbb{K}\) in the usual sense and that ( ∗ ) fails to hold. Then \(g \in \mathbb{K}[x]\) exists with degg < degf and \(g(a) = f(a) = 0\). But \((f,g)_{\mathbb{K}} = 1\), so \(p,q \in \mathbb{K}[x]\) exist such that \(p(x)f(x) + q(x)g(x) = 1\), and setting x = a implies 0 = 1. Conversely, if f satisfies ( ∗ ), it cannot be reducible, for then f(x) = g(x)h(x), where g, h have degrees less than n. If \(a \in \mathbb{C}\) is a root of f, then \(0 = f(a) = g(a)h(a)\) implies without loss of generality that g(a) = 0, contrary to ( ∗ ).
- 23.
- 24.
See Table 3.2 in Gray’s book [255, p. 87].
- 25.
Chapter III of Gray’s book [255] is devoted to all the work done on Fuchs-type equations that can be integrated algebraically.
- 26.
Stenographic notes of these lectures were reproduced by the university. Copies are located in the Bibliothek Mathematik und Geschichte der Naturwissenschaften at the University of Hamburg.
- 27.
Weierstrass’ negative opinion of Frobenius’ fiancée and of his decision to leave Berlin for Zurich is contained in a letter to Sonya Kovalevskaya dated 23 September 1875 [28, p. 219].
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Hawkins, T. (2013). A Berlin Education. 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_1
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