Introduction to the Theory of Algebraic Equations (Classic Reprint)

The solution of the general quadratic equation was known as early as the ninth century; that of the general cubic and quartic equations was discovered in the sixteenth century. During the succeeding two centuries many imsuccessful attempts were made to solve the general equations of the fifth and higher degrees. In 1770 Lagrange analyzed the methods of I iis predecessors and traced all their results to one principle, that of rational resolvents, and proved that the general quintic equation cannot be solved hy rational resolvents. The impossibility of the algebraic solution of the general equation of degree nfn 4), whether by rational or irrational resolvents, was then proved by A bel, Wantzel, and Galois. Out of these algebraic investigations grew the theory of substitutions and groups. The first systematic study of substitutions was made by Cauchy {J ournal de VS cole poly technique, 1815). The subject is here presented in the historical order of its development. The First Part (pp. 1-41) is devoted to theL agrangeC auchy-A bel theory of general algebraic equations. The Second Part (pp. 42-98) is devoted toG alois theory of algebraic equations, whether with arbitrary or special coefficients. The aim has been to make the presentation strictly elementary, with practically no dependence upon any branch of mathematics beyond elementary algebra. There occiu niunerous illustrative examples, as well as sets of elementary exercises.
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