Arithmetic Investigations: Groups

Abstract

AbstractFrobenius’ interest in arithmetic problems during the late 1870s was not limited to bilinear forms. The groundbreaking work of Gauss on the composition of binary quadratic forms had brought with it a line of thinking that would now be characterized as group-theoretic. This same line of thought resurfaced in Kummer’s revolutionary work on his theory of ideal numbers and prompted Ernst Schering to develop it into what would now be interpreted as the existence part of the fundamental theorem of finite abelian groups, already expressed ambiguously by Schering so as to encompass the finite abelian groups that were implicit in Gauss’ theory of composition of binary quadratic forms as well as those in Kummer’s theory of ideal numbers (ideal class groups). Soon thereafter, both Kronecker and Dedekind expressed Schering’s result explicitly in abstract terms, with Dedekind expressly making the connection with Galois’ notion of a group.KeywordsConjugacy ClassGalois GroupCyclic SubgroupDensity TheoremDouble CosetThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.