Linear groups in Galois fields. A case study of tacit circulation of explicit knowledge

TL;DR

This paper examines the historical circulation of explicit and tacit algebraic knowledge among French and American mathematicians around the early 20th century, focusing on the development of finite group theory and the role of linear groups in Galois fields.

Contribution

It provides a case study of how shared algebraic cultures and knowledge circulation influenced the development of finite group theory in the early 20th century.

Findings

Identifies the role of collective algebraic practices in shaping mathematical culture.

Highlights the influence of French-American mathematical exchanges on group theory.

Shows the importance of tacit knowledge in the development of algebraic concepts.

Abstract

This preprint is the extended version of a paper that will be published in the proceedings of the Oberwolfach conference "Explicit vs tacit knowledge in mathematics" (January 2012). It presents a case study on some algebraic researches at the turn of the twentieth century that involved mainly French and American authors. By investigating the collective dimensions of these works, this paper sheds light on the tension between the tacit and the explicit in the ways some groups of texts hold together, thereby constituting some shared algebraic cultures. Although prominent algebraists such as Dickson made extensive references to papers published in France, and despite the roles played by algebra and arithmetic in the development of the American mathematical community, our knowledge of the circulations of knowledge between France and the United States at the beginning of the 20th century is still very limited. It is my aim to tackle such issues through the case study of a specific collective approach to finite group theory at the turn of the 20th century. This specific approach can be understood as a shared algebraic culture based on the long run circulation of some specific procedures of decompositions of the analytic forms of substitutions. In this context, the general linear group was introduced as the maximal group in which an elementary abelian group (i.e., the multiplicative group of a Galois field) is a normal subgroup.