Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

~ 159] RESOLVENT EQUATIONS AND THEIR GROUPS 303

For the domain R' = (R, ), the group of the cubic equation
is the cyclic group C3 (~ 156). The substitution (XIX2X3) of
C3 replaces the functions
4 /= Xl + X2 + -2X3, X=Xl +- 2X2 + WX3,
with coefficients in R, by w24 and ox, respectively. Thus
the substitutions of C3 leave,3 and x3 unaltered. Hence
(~ 155) the latter are rational functions, with coefficients in
R, of 6. We have
f,3+x3=a=2c13-9Clc2+27c3, 4,3_X3=-3/-3^,
4,3 = (a- 3V/- 3).
A cube root 4\ of the last quantity is adjoined to R'; the
group is thereby reduced to the identity group G1 to which 4
belongs. The roots xi are now in the enlarged domain (R', 4).
From the expressions for cl, 4, x, we find by multiplications
X1 = ci+l+X), X2 =I (Cl +24+X), X3 =Cl3 CL w+W2x).
Here X= (C2-3c2)/4. In brief, the above solution consists
in finding, by means of a quadratic equation, a function,3
which belongs to C3, and then finding, by means of a binomial
cubic equation, the function 4 which belongs to G1.
Taking cli=0, we obtain Cardan's formulas (~ 157).
159. Resolvent Equations and their Groups. The auxiliary
quadratic and binomial cubic equations employed in the solution * of the general cubic equation are called resolvent equations of the latter. In general, let f(x) =0 be any given equation with coefficients in a given domain R, and let 4 be a rational
function of its roots with coefficients in R. If 4 belongs to a
subgroup H of index v under the group G of f(x)=0 for R,
we have seen (~ 154) that 4 is a root of a resolvent equation
of degree v with coefficients in R. Suppose that we can solve
this resolvent equation relatively to R. By adjoining its
root 4 to the domain R, we obtain a domain Ri=(R, 4,) for
which the group of f(x) =0 is H. If repeated adjunctions
lead to a domain R, for which the group is the identity G1,
* For brevity, we omit the words " by radicals " after "solution " or "solve."