786] 
EQUATION. 
519 
becomes reducible; for instance, the general quadric equation x 2 + px + q = 0 is irre 
ducible, but it becomes reducible, breaking up into rational linear factors, when we 
adjoin the radical J\p~ — q- 
The fundamental theorem is the Proposition I. of the “ Memoire sur les conditions 
de resolubilitd des equations par radicaux ”; viz. given an equation of which a, b, c,... 
are the m roots, there is always a group of permutations of the letters a, b, c,... 
possessed of the following properties:— 
1. Every function of the roots invariable by the substitutions of the group is 
rationally known. 
2. Reciprocally, every rationally determinable function of the roots is invariable 
by the substitutions of the group. 
Here by an invariable function is meant not only a function of which the form is 
invariable by the substitutions of the group, but further, one of which the value is 
invariable by these substitutions: for instance, if the equation be (f>x = 0, then cf)X is 
a function of the roots invariable by any substitution whatever. And in saying that 
a function is rationally known, it is meant that its value is expressible rationally in 
terms of the coefficients and of the adjoint quantities. 
For instance, in the case of a general equation, the group is simply the system of 
the 1.2.3 ... n permutations of all the roots, since, in this case, the only rationally 
determinable functions are the symmetric functions of the roots. 
In the case of the equation x n ~ l + ...+«+ 1 = 0, n a prime number, 
a, b, c, ...,k = r, r9, ro\ ..., r9 n 2 , 
where g is a prime root of n, then the group is the cyclical group abc ... k, 
be... ka, ..., kab ...j, that is, in this particular case the number of the permutations 
of the group is equal to the order of the equation. 
This notion of the group of the original equation, or of the group of the equation 
as varied by the adjunction of a series of radicals, seems to be the fundamental one 
in Galois’s theory. But the problem of solution by radicals, instead of being the 
sole object of the theory, appears as the first link of a long chain of questions relating 
to the transformation and classification of irrationals. 
Returning to the question of solution by radicals, it will be readily understood 
that by the adjunction of a radical the group may be diminished; for instance, in 
the case of the general cubic, where the group is that of the six permutations, by 
the adjunction of the square root which enters into the solution, the group is reduced 
to abc, bca, cab; that is, it becomes possible to express rationally, in terms of the 
coefficients and of the adjoint square root, any function such as a 2 b + b-c + c 2 a which 
is not altered by the cyclical substitution a into b, b into c, c into a. And hence, 
to determine whether an equation of a given form is solvable by radicals, the course 
of investigation is to inquire whether, by the successive adjunction of radicals, it is