Leonard Eugene Dickson and his work in the arithmetics of algebras
What is the appropriate generalization of the concept of integer for general algebras? This paper describes the historical and mathematical context for Dickson's general formulation in a way that touches upon aesthetic and social factors leading to the acceptance of a theory. In 1801, Gauss studied complex integers and their divisibility properties. Later, Ernst Kummer investigated more general algebraic integers and was dismayed at the failure of unique factorization. Rudolf Lipschitz in 1886 and Adolf Hurwitz in 1896 had competing definitions for quaternionic integers, which were complicated by the noncommutativity of multiplication. Hurwitz, already expert in the theory of algebraic numbers, brought more sophistication in relating the integers to the whole algebra. In the first two decades of the twentieth century, his student, Gustave du Pasquier extended the investigation first to matrices and then more generally. His struggles yielded some fruit, but his formulation lacked unity and ability to deal with unique factorization. This was left to Leonard Eugene Dickson, who in the early 1920s, began with quaternions and ended up with a general theory of hypercomplex numbers that recognized the significance of the structure theorems of Wedderburn.
scientific article; zbMATH DE number 1139962