By Israel Kleiner
This ebook does not anything lower than supply an account of the highbrow lineage of summary algebra. the improvement of summary algebra used to be propelled by way of the necessity for brand new instruments to handle yes classical difficulties that seemed insoluble by way of classical potential. a massive subject of the ebook is to teach how summary algebra has arisen in trying to resolve a few of these classical difficulties, delivering a context from which the reader may perhaps achieve a deeper appreciation of the maths concerned. arithmetic teachers, algebraists, and historians of technology will locate the paintings a priceless reference.
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Additional info for A History of Abstract Algebra
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 .
Xn ) assumes under all the n! permutations of the roots x1 , x2 , x3 , . . , xn of f (x). If these are denoted by y1 , y2 , y3 , . . , 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).
We now give a sample of some implicit group-theoretic work in number theory, especially in algebraic number theory. 2). Algebraic number fields and their arithmetical properties were the main objects of study. In 1846 Dirichlet studied the units in an algebraic number field and established that (in our terminology) the group of these units is a direct product of a finite cyclic group and a free abelian group of finite rank. At about the same time Kummer introduced his “ideal numbers,” defined an equivalence relation on them, and derived, for cyclotomic fields, certain special properties of the number of equivalence classes, the so-called class number of a cyclotomic field—in our terminology, the order of the ideal class group of the cyclotomic field.
A History of Abstract Algebra by Israel Kleiner