Goldstein / Schappacher / Schwermer The Shaping of Arithmetic after C.F. Gauss's Disquisitiones Arithmeticae

1. Auflage 2007
ISBN: 978-3-540-34720-0
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 578 Seiten, eBook

Since its publication, C.F. Gauss's Disquisitiones Arithmeticae (1801) has acquired an almost mythical reputation, standing as an ideal of exposition in notation, problems and methods; as a model of organisation and theory building; and as a source of mathematical inspiration. Eighteen authors - mathematicians, historians, philosophers - have collaborated in this volume to assess the impact of the Disquisitiones, in the two centuries since its publication.

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.

Zielgruppe

Research

Autoren/Hrsg.

Goldstein, Catherine

Schappacher, Norbert

Schwermer, Joachim

Inhaltsverzeichnis

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5.4;II.4 The Unpublished Section Eight: On the Way to Function Fields over a Finite Field;167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

Über Autor(innen)

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.