Abstract
During the decade spanning his return from Italy and his departure for Göttingen, twenty-one publications by Dirichlet appeared. Discounting translations and excerpts, these represent thirteen separate contributions. They differ from his previous work in several respects.
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Notes
- 1.
Kronecker 1888b; see his Math. Werke 5:476.
- 2.
See Werke 2:6.
- 3.
Kronecker 1882; see Kronecker Werke 2:354. Also see the opening of Kirchhoff 1876, the Mechanik.
- 4.
For Poinsot’s addition and for Dirichlet’s 1853c, see Lagrange 1853:389–98 and 399–401, respectively; for Bertrand’s note on Dirichlet’s rigorous simplification of Lagrange’s proof, see Lagrange 1853:64.
- 5.
Werke 1:641fn.
- 6.
Werke 1:642.
- 7.
Werke 1:644.
- 8.
Liouville also had published a French translation of Gauss’s memoir [Gauss 1840] in his Journal in 1842.
- 9.
Encke 1862. In this comment, Encke used the opportunity not only to mention to members of the Akademie his own relationship to Gauss but to explain that Gauss’s interest in Seeber’s work was consistent with Gauss’s [Shakespearean] motto “Thou Nature art my goddess, to thy law / My services are bound.”
- 10.
Gauss 1831; see Gauss Werke 2:191.
- 11.
For details, see Werke 2:24–26.
- 12.
Minkowski 1891; see his Ges. Abh. 1:244–45.
- 13.
In addition to references we have cited that Minkowski made to Dirichlet on other occasions, it is of interest to note how much emphasis he placed on Dirichlet’s work in the paper [Minkowski 1896] presented at the 1893 Chicago Congress. Dirichlet’s influence on Minkowski was to be of considerable importance for another direction of twentieth-century number theory. Although a more extended discussion of the impact of 1850b would lead us beyond the chronological boundary of our volume, it is worth noting that it was this memoir, along with 1839a and Kurt Hensel’s p-adic number theory, which, after 1920, would guide Helmut Hasse to the Hasse–Minkowski theorem and his “global-local principle.”
- 14.
See the D.A., art. 304, and the last item (Sect. 13.13) in this chapter.
- 15.
See Dirichlet’s concluding remarks of 1839–1840. Legendre’s initial attempts at a proof concerning the sum of three squares are found in [Legendre 1788], which, as André Weil noted in Weil 1983:331–32, may have interested Gauss in the problem.
- 16.
As noted previously, Gauss later in the D.A. and in subsequent work simply referred to it as the Fundamental Theorem.
- 17.
This is explained by the fact that this proof, included in part I of the D.A., belongs to the original version of the D.A. intended to be a work on congruences. Gauss’s change in orientation occurred only with completion of his work on Part V of the D.A. See Merzbach 1981.
- 18.
Dirichlet gave a reference to his 1842b and to art. 193 of the D.A. To avoid repetition, note our Chapter 16, where we outline the relevant discussion as found in the editions of Dirichlet–Dedekind’s number theory lectures, especially paragraphs 82, etc.
- 19.
Werke 2:180–81.
- 20.
Gauss, D.A., art. 304.
- 21.
Werke 1:472.
- 22.
For Euler’s related efforts, see Weil 1983:224–26.
- 23.
Attempts to settle the questions Gauss raised in articles 303 and 304 of the D.A. have occupied mathematicians to our times. For example, in 2007, H. M. Stark, who resolved several of these issues, in the introduction to his careful discussion of “The Gauss Class-Number Problems” called attention to the conjectures of Gauss’s articles 303 and 304; to the impact of Heilbronn 1934 (“[Gauss’s] addendum [in art. 304], caused much heartache when in 1934 Heilbronn finally proved that k(d) approaches \(\infty \) as d approaches \(-\infty \) ineffectively.”); to expanded interpretations; and to related open questions. See Stark 2007:247–50.
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Merzbach, U.C. (2018). Publications: 1846–1855. In: Dirichlet. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-01073-7_13
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