Self-portraits with Évariste Galois (and the shadow of Camille Jordan)

This paper investigates the collections of 19th century texts in which Evariste Galois's works were referred to in connection to those of Camille Jordan. Before the 1890s, when object-oriented disciplines developed, most of the papers referring to Galois have underlying them three main networks of texts. These groups of texts were revolving around the works of individuals: Kronecker, Klein, and Dickson. Even though they were mainly active for short periods of no more than a decade, the three networks were based in turn on specific references to the works of Galois that occurred in the course of the 19th century. By questioning how mathematicians were portraying themselves and their mathematics through their references to Galois, this paper therefore sheds new light on some collective interpretations of Galois's works. It especially highlights the important role played in the long term legacy of Galois by some practices of reduction modeled on the analytic representation of the decomposition of linear substitutions into two forms of actions of cycles. Complementary to the local study of these networks, the article proposes a more global analysis. Galois's works were often related to the problem of the "classification and transformation" of the "irrationals." Contrary to what has become, in the 20th century, a commonplace of the historiography of algebra, and distinct from the teaching of courses in Algèbre supérieure, Galois's works were fitted into classifications of mathematical knowledge neither under the heading of the theory of equations nor as part of the theory of substitutions. For most of the 19th century, the problem of the irrationals involved elliptic (or abelian) functions (and therefore complex analysis). The impossibility of solving general algebraic equations of degree greater than four by radicals highlighted the necessity of characterizing the special nature of the irrational quantities and functions defined by both algebraic and differential equations.