The Shaping of Arithmetic after C.F. Gauss's Disquisitiones Arithmeticae (eBook)

Catherine Goldstein is Directrice de recherches du CNRS and works at the Institut de mathématiques de Jussieu (Paris, France). She is the author of "Un théorème de Fermat et ses lecteurs" (1995) and a coeditor  of "Mathematical Europe: History, Myth, Identity"(1996). Her research aims at developing a social history of mathematical practices and results, combining close readings and a network analysis of texts. Her current projects include the study of mathematical sciences through World War I and of experimentation in XVII th-century number theory.Norbert Schappacher is professor of mathematics at Université Louis Pasteur, Strasbourg.His mathematical interests relate to the arithmetic of elliptic curves.But his current research projects lie in the history of mathematics. Specifically, he focuses on the intertwinement of philosophical and political categories with major junctures in the development of mathematical disciplines in the XIX/up{th} and XX/up{th} centuries. Examples include number theory and algebraic geometry, but also medical statistics. Joachim Schwermer is professor of mathematics at University of Vienna. In addition, he serves as scientific director at the Erwin-Schroedinger International Institute for Mathematical Physics, Vienna. His research interests lie in number theory and algebra, in particular, in questions arising in arithmetic algebraic geometry and the theory of automorphic forms. He takes a keen interest in the mathematical sciences in the XIX/up{th} and XX/up{th} centuries in their historical context.

Foreword 6
Table of Contents 8
Editions of Carl Friedrich Gauss’s Disquisitiones Arithmeticae 10
Part I A Book’s History 12
I.1 A Book in Search of a Discipline ( 1801 - 1860) 14
I.2 Several Disciplines and a Book ( 1860 - 1901) 78
Part II Algebraic Equations, Quadratic Forms, Higher Congruences: Key Mathematical Techniques of the Disquisitiones Arithmeticae 115
II.1 The Disquisitiones Arithmeticae and the Theory of Equations 116
II.2 Composition of Binary Quadratic Forms and the Foundations of Mathematics 137
II.3 Composition of Quadratic Forms: An Algebraic Perspective 153
II.4 The Unpublished Section Eight: On the Way to Function Fields over a Finite Field 167
Part III The German Reception of the Disquisitiones Arithmeticae: Institutions and Ideas 207
III.1 A Network of Scientific Philanthropy: Humboldt’s Relations with Number Theorists 208
III.2 The Rise of Pure Mathematics as Arithmetic with Gauss 242
Part IV Complex Numbers and Complex Functions in Arithmetic 276
IV.1 From Reciprocity Laws to Ideal Numbers: An ( Un) Known Manuscript by E. E. Kummer 278
IV.2 Elliptic Functions and Arithmetic 298
Part V Numbers as Model Objects of Mathematics 320
V.1 The Concept of Number from Gauss to Kronecker 322
V.2 On Arithmetization 350
Part VI Number Theory and the Disquisitiones in France after 1850 382
VI.1The Hermitian Form of Reading the Disquisitiones 384
VI.2 Number Theory at the Association française pour l’avancement des sciences 418
Part VII Spotlighting Some Later Reactions 435
VII.1 An Overview on Italian Arithmetic after the Disquisitiones Arithmeticae 436
VII.2 Zolotarev’s Theory of Algebraic Numbers 458
VII.3 Gauss Goes West: The Reception of the Disquisitiones Arithmeticae in the USA 468
Part VIII Gauss’s Theorems in the Long Run: Three Case Studies 485
VIII.1 Reduction Theory of Quadratic Forms:Towards Räumliche Anschauung in Minkowski’s Early Work 486
VIII.2 Gauss Sums 508
VIII.3 The Development of the Principal Genus Theorem 532
Table of Illustrations 565
Index 567
Authors’ addresses 578