Leonard Eugene Dickson and his work in the arithmetics of algebras
type: Item
instance of: scholarly article
title: Leonard Eugene Dickson and his work in the arithmetics of algebras
zbMATH Open document ID: 0987.01016
author: Della Dumbaugh Fenster
published in: Archive for History of Exact Sciences
publication date: 2002-06-19T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 01A70
Mathematics Subject Classification ID: 01A60
Mathematics Subject Classification ID: 01A55
Mathematics Subject Classification ID: 16-03
zbMATH DE Number: 1139962
zbMATH Keywords: algebras
zbMATH Keywords: integers
zbMATH Keywords: L. E. Dickson
reviewed by: Edward J. Barbeau
MaRDI profile type: Publication
full work available at URL: s004070050014
OpenAlex ID: W2109049749
DOI: 10.1007/S004070050014
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description: scientific article; zbMATH DE number 1139962