The Evolution of Algebra 1800-1870

Abstract

The first event of this period was the appearanceS in 1801, of C. F. Gauss' Disquisitiones Arithmeticae. Of the seven parts of the book only one is devoted to an algebraic issue, namely the cyclotomic equation xn 1 = 0. But the authorns brilliant algebraic thinking is apparent in all the other parts as well. Disquzsitionesn an epoch-making work in algebraic number theory, was for a long time a handbook and source of ideas in algebra. In the course of his study of the cyclotomic equation Gauss shows that it is solvable for every n in the sense that the solutions are expressible in terms of radicals, gives a method for explicitly finding these expressions7 and singles out the values of n for which the solutions are expressible in quadratic radicals and thus the values of n for which it is possible to construct a regular n-gon by means of ruler and compass. As always7 his investigations are strikingly profound and detailed. They were continued by N. H. Abel? who proved the insolvability by radicals of the general quintic and singled out a class of equations7 now named for him7 that are solvable by radicals. The new notions of field (domain of rationality) and group (group of an equation) turned up in Abel's papers with greater definiteness. The next step in this direction that cornpleted the theory was the papers of the young E. Galois, published in fragmentary form between 1830 and 1832, and, after his death, in more complete form by Liouville in 1846. The papers of Abel7 and especially of Galois, already belong to the radically new trend of ideas now generally accepted in algebra. In his study of the ancient problem of solution of equations by radicals Galois shifted the center of gravity from the problem to the methods of its solution: he gave clear-cut definitions of the concepts of a field and of the group of an equation7 established the correspondence between the subgroups of the group of an equation and the subfields of the splitting field of the polynomial on the left side of that equation, and, finally, singled out the normal subgroups of a group and studied its composition series. These were completely new and extremely fruitful methods of investigation and yet