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Algebra without “all the fictions about the roots of equations”: the theory of characteristics in Kronecker’s Vorlesungen über die algebraischen Gleichungen. (L’algèbre sans “les fictions des racines” : Kronecker et la théorie des caractéristiques dans les Vorlesungen über die algebraischen Gleichungen.) (French. English summary) Zbl 1447.01012

Leopold Kronecker’s lectures on the theory of algebraic equations in Berlin from 1872 to 1891 are documented by course notes taken by his students that are now at the mathematics library of the University of Strasbourg [http://docnum.unistra.fr/digital/collection/coll7]. This article presents part of the author’s PhD thesis [La théorie des caractéristiques dans les Vorlesungen über die Theorie der algebraischen Gleichungen de Kronecker : la fin du cycle d’idées sturmiennes? Paris: Université Sorbonne Paris Cité (2017), http://www.theses.fr/2017USPCC238]. It provides a general introduction to these lectures with the goal of “grasping some essential notions, of finding their origin, of following their evolution and transformation” as P. Dugac [Arch. Hist. Exact Sci. 10, 41–176 (1973; Zbl 0259.01012)] did for Karl Weierstrass. In these, Sturm’s theorem on the number of real roots of a given polynomial in a given interval (see [Hourya Benis Sinaceur, in: Collected works of Charles-François Sturm. Basel: Birkhäuser, 13–24 (2009; Zbl 1149.01023)]) plays an important rôle.
In all lectures but the last one, Kronecker gives a proof of its generalisation by James Joseph Sylvester [in: The collected mathematical papers. Vol. I (1837–1853). Cambridge: Cambridge University Press (1904; JFM 35.0020.01), p. 40–46, 429–586] that relies on the “geometric intuition” of the intermediate value theorem. Then, in the lectures until 1881 and after 1888, he proposes yet another generalisation, for the investigation of which he introduces the concept of characteristic of a system of polynomial functions. For two functions \(\varphi,\psi\), Kronecker provides it with “the following geometric meaning: the characteristic \(\chi(\varphi,\psi)\) is half the excess of the exits over the entrances [on the \(z\)-axis w.r.t.the space between the curves \(y=\varphi(z)\) and \(y=\psi(z)\)] when we move on the curve \(y=\varphi(z)\), i.e.count the [real roots of \(\varphi\)], and equal to half the excess of the entrances over the exits when we move on the curve \(y=\psi(z)\), i.e.count the [real roots of \(\psi\)]”. His motivation is that “the core of the first and fourth proof of [the fundamental theorem of algebra] by Gauss (1799 and 1849, respectively) is at bottom nothing else than the determination of the characteristic”. This article presents this material based on the course notes dated 1880–1881 and 1890–1891 (from which the citations above are taken, respectively), and inspects Kronecker’s reflections on the concept of root of an algebraic equation, which emphasise the importance of separating real roots. He also discusses Kronecker’s “Fortgangsprinzip” for orienting curves.
The reviewer has a few comments to add. On p. 8, it would be better to write that Kronecker is granted the right to bear the title of professor in 1864 rather than that he “becomes Prädikat Professor”, so that the reader grasps that this does not imply a university position. On pp. 38 and 83, there is an issue of anachronism with respect to the concept of real number as ideated by Richard Dedekind [Stetigkeit und irrationale Zahlen. Braunschweig: F. Vieweg u.Sohn (1872)] when the author writes that “Bolzano’s proof must therefore use the ‘completeness’ of the field of reals” and, for Gauss’s second proof of the fundamental theorem of algebra [“Demonstratio nova altera theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse”, Comment. Soc. Reg. Sci. Gott. Recent. 3, 107–133 (1816), https://www.paultaylor.eu/misc/gauss-web], that “one knows that a proof that would not take into account, even remotely, the construction of the reals, would have little chance of success” with a subsequent evocation of \(\mathbb{C}[x]\): in fact, in both cases, the proofs rely on properties of a magnitude whose truth may be considered more reliable than the construction of the reals, which is based on the concept of the set of all subsets of the set of rational numbers.
On p. 45, line 3, read “(1)”, not “(2)”; on the same page below, the author could specify that \(g_h\) is introduced as the quotient of \(f_{h-1}\) by \(f_h\) and that \(\xi_h\) is introduced as a root of \(f_h\). On p. 49, note 113 should define a Jordan curve as the homeomorphic image of a circle. On p. 50, note 116, read “3. Teil”, not “z.Teil”. On p. 74, note 141, read “64. Stück”, not “64. März”. On p. 77, the author writes that in his lecture in the winter term of 1886–1887, Kronecker uses symmetric polynomials in the roots of a given polynomial to show that all its real roots lie in a certain interval \({-g}\mathrel{\dots}{+g}\); in fact, he makes the usual estimation to compute this interval.

MSC:

01A55 History of mathematics in the 19th century
01A70 Biographies, obituaries, personalia, bibliographies
12-03 History of field theory

Biographic References:

Kronecker, Leopold
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