A History of Galois fields

Frederic Brechenmacher (1, 2)

1 LML - Laboratoire de Mathématiques de Lens
2 X-DEP-HSS - Département d'Humanités et Sciences Sociales de l'École polytechnique

This paper stresses a specific line of development of the notion of finite field, from Évariste Galois's 1830 ''Note sur la théorie des nombres,'' and Camille Jordan's 1870 Traité des substitutions et des équations algébriques, to Leonard Dickson's 1901 Linear groups with an exposition of the Galois theory. This line of development highlights the key role played by some specific algebraic procedures. These intrisically interlaced the indexations provided by Galois's number-theoretic imaginaries with decompositions of the analytic representations of linear substitutions. Moreover, these procedures shed light on a key aspect of Galois's works that had received little attention until now.

Mots clés: substitutions; group theory; equations; Galois; Jordan; Hermite; Dickson; Moore; history of algebra; history of number theory; Galois fields; finite fields; cyclotomy

Domaines: Histoire, Philosophie et Sociologie des sciences

Frederic Brechenmacher. A History of Galois fields. 2012. ⟨hal-00786662⟩