788] 
543 
788. 
GALOIS. 
[From the Encyclopaedia Britannica, Ninth Edition, voi. x. (1879), p. 48.] 
Galois, Evariste (1811—1832), an eminently original and profound French mathe 
matician, born 26th October 1811, killed in a duel May 1832. A necrological notice 
by his friend M. Auguste Chevalier appeared in the Revue Encyclopédique, September 
1832, p. 744 ; and his collected works are published, Liouville, t. XL (1846), pp. 381— 
444, about fifty of these pages being occupied by researches on the resolubility of 
algebraic equations by radicals. But these researches, crowning as it were the previous 
labours of Lagrange, Gauss, and Abel, have in a signal manner advanced the theory, 
and it is not too much to say that they are the foundation of all that has since 
been done, or is doing, in the subject. The fundamental notion consists in the 
establishment of a group of permutations of the roots of an equation, such that every 
function of the roots invariable by the substitutions of the group is rationally known, 
and reciprocally that every rationally determinable function of the roots is invariable 
by the substitutions of the group; some further explanation of the theorem, and in 
connexion with it an explanation of the notion of an adjoint radical, is given under 
Equation, No. 32, [786]. As part of the theory (but the investigation has a very high 
independent value as regards the Theory of Numbers, to which it properly belongs), 
Galois introduces the notion of the imaginary roots of an irreducible congruence of a 
degree superior to unity ; i.e., such a congruence, F (x) = 0 (mod. a prime number p), 
has no integer root ; but what is done is to introduce a quantity i subjected to the 
condition of verifying the congruence in question, F (i) = 1 (mod. p), which quantity 
i is an imaginary of an entirely new kind, occupying in the theory of numbers a 
position analogous to that of J — 1 in algebra.