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Article ID: CBB001211763

Cyclotomie et formes quadratiques dans l'œuvre arithmétique d'Augustin-Louis Cauchy (1829--1840) (2013)

Boucard, Jenny (Author)

Archive for History of Exact Sciences
Volume: 67
Pages: 1--66

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Arithmetic Mathematics Algebra Number theory; number concept 19th century Cauchy, Augustin Louis

Augustin-Louis Cauchy publie une majorité de ses recherches arithmétiques entre 1829 et 1840. Celles-ci ne sont pourtant qu'évoquées dans certaines histoires de la théorie des nombres centrées sur les lois de réciprocité ou sur la théorie des nombres algébriques. Elles y sont décrites comme contenant quelques résultats similaires à ceux de Gauss, Jacobi ou Dirichlet mais de manière incomplète et désordonnée. L'objectif de cet article est de présenter une analyse des textes arithmétiques de Cauchy publiés entre 1829 et 1840 pour montrer qu'ils contiennent au contraire un ensemble cohérent de résultats en lien avec les formes quadratiques 4pmu=x2+ny2 , où p est un nombre premier et n un diviseur de p-1 . Nous discuterons également la forme particulière de ce corpus et la stratégie utilisée pour retrouver les lignes directrices du travail de Cauchy. Augustin-Louis Cauchy published most of his arithmetical research between 1829 and 1840. These are however only mentioned in some number theory history centered on reciprocity laws or on theory of algebraic numbers. They are described as containing some results similar to those of Gauss, Jacobi and Dirichlet but in a incomplete and disorganized way. The objective of this paper is to present an analysis of Cauchy's arithmetical texts published between 1829 and 1840 to show that they contain a rather consistent set of results related to quadratic forms 4pmu=x2+ny2 , where p is a prime and n a divisor of p-1 . We will also discuss the particular form of this body of texts and the strategy we used to find the guidelines of the work of Cauchy.

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