By Israel Kleiner
This publication does not anything below supply an account of the highbrow lineage of summary algebra. the advance of summary algebra used to be propelled by way of the necessity for brand spanking new instruments to handle yes classical difficulties that seemed insoluble by way of classical ability. an immense subject matter of the ebook is to teach how summary algebra has arisen in trying to clear up a few of these classical difficulties, supplying a context from which the reader may well achieve a deeper appreciation of the maths concerned. arithmetic teachers, algebraists, and historians of technological know-how will locate the paintings a useful reference.
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Extra resources for A History of Abstract Algebra
In symbols, θr θs = θt . II. It is always true that (θr θs )θt = θr (θs θt ) = θr θs θt . III. From θ θr = θ θs or from θr θ = θs θ it follows that θr = θs . Weber’s and other definitions of abstract groups given at the time applied only to finite groups. They thus encompassed the two theories of permutation groups and (finite) abelian groups, which derived from the two sources of classical algebra— polynomial equations and number theory, respectively. Infinite groups, which arose from the theories of (discontinuous and continuous) transformation groups, were not subsumed under those definitions.
Yk , then the resolvent equation is given by g(x) = (x − y1 )(x − y2 ) · · · (x − yk ). It is important to note that the coefficients of g(x) are symmetric functions in x1 , x2 , x3 , . . , xn , hence they are polynomials in the elementary symmetric functions of x1 , x2 , x3 , . . , xn ; that is, they are polynomials in the coefficients of the original equation f (x). —the source of what we call Lagrange’s theorem in group theory. For example, if f (x) is a quartic with roots x1 , x2 , x3 , x4 , then R(x1 , x2 , x3 , x4 ) may be taken to be x1 x2 + x3 x4 , and this function assumes three distinct values under the twenty-four permutations of x1 , x2 , x3 , x4 .
The earliest monograph devoted entirely to abstract groups was the book by J. A. de Séguier of 1904 entitled Elements of the Theory of Abstract Groups . At the very beginning of the book there is a set-theoretic introduction based on the work of Cantor: “De Séguier may have been the first algebraist to take note of Cantor’s discovery of uncountable cardinalities” . Next is the introduction of the concept of a semigroup with two-sided cancellation law and a proof that a finite semigroup is a group.
A History of Abstract Algebra by Israel Kleiner