EQUATIONS, Galois' Theory of.— In the )6th century the Italian mathematicians suc ceeded in solving the cubic and biquadratic equations. Their brilliant achievements must have made it seem probable that the solution of the equations of fifth and higher degrees would soon be found. Such, however, was not to be the case. For two centuries the first mathematicians of the day essayed in vain to solve the quintic. Tschimhausen, Euler, Van dermonde, Malfatti and Lagrange embodied their researches in valuable memoirs, but at the dose of the 18th century the solution of the equation of the fifth degree seems farther away than ever.
In their apparent defeat, however, lay the germs of ultimate victory. As a result of all these investigations it became manifest that the solution of algebraic equations and certain groups of substitutions of their roots were inti mately related. In the case of the general equations of degrees three and four this rela tion was very clear indeed; it was less clear in regard to the general equation of degree n, and still more hazy in regard to the special equa tions which had been considered up to that time. It was reserved to Evareste Galois to put these loose ends together and to develop a theory of the solution of algebraic equations at once simple and far-reaching. Indeed. the ideas of Galois are not only fundamental in most algebraic investigations, but they have also been extended by Lie and others with great effect to the theory of differential equations. But even
here they do not stop. It is in Galois' theory that the notion of a group first came promi nently before the mathematical public; a notion which to-day pervades a good part of the whole domain of mathematics.
Galois died at the age of 22 (1832). Twice he presented memoirs to the Paris Acad emy, containing an account of his theory. The first was lost, the second was returned to its youthful author by Poisson as unintelligible. Galois' theory was first made public to the mathematical world in 1846 when Liouville published this latter memoir without comments. In 1858 Betti published an exposition of Galois' theory with complete proofs and some valuable extensions.
Lagrange in his great memoir of 1770-71 developed what he styled a calcul des combina tions and which is in fact the origin of Galois' Theory of Equations. This new calcul was further developed in a number of papers by Ruffini, beginning 1799, who tried to demonstrate the impossibility of the algebraic solution of the general equation of degrees greater than four by th's means; by Gauss (1801) and Lagrange (1808) in the solution of the equations on which the roots of unity depend; and finally by Abel, who, besides being the first to rigorously demon strate the insolvability of the quintic by radi cals (1826), discovered a new class of algebraic solvable equations which occur in the division of the elliptic functions (1829).