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

302

EQUATIONS SOLVABLE BY RADICALS

[CH. XV

and root extractions performed upon its coefficients as well as
upon the coefficients and roots of the preceding equations
of the series. IXi order to focus our attention upon a particular
equation of a series, and to have a general and hence simpler
phraseology, it is convenient to have the following generalization of the above definition of solvability by radicals.
An equation with coefficients in a domain R(ki,..., km)
shall be said to be solvable by radicals relatively to R if all of
its roots can be derived by rational operations and root extractions performed upon ki,..., km or upon quantities obtained
from them by those operations.
For example, xl3=213 is evidently solvable relatively to R(e), where
e is a particular imaginary 13th root of unity.
While a quintic equation whose coefficients ci,..., c5 are independent
variables will be shown to be not solvable by radicals, i.e., relatively to
R(ci,..., c5), it is solvable relatively to R'=R(xl, cl,..., 5), where
xl is one root of the quintic (since its group for R' is a solvable group of
order 24). We have merely shifted the difficulty to the determination
of the new domain R'. The benefit that may be gained by the use of R'
is merely one of phraseology.
158. Solution of a Cubic Equation. In
X3-ClX2-+C2X-C3 =0,
let Cl, C2, C3 be independent variables. This general cubic
equation will be discussed from the group standpoint with
the aim of providing a concrete illustration of the general
theory which is to follow.
Let w be an imaginary cube root of unity. For the domain
R=R(w, C1, C2, ca), the group of the cubic equation is the
symmetric group G6 on the roots xl, X2, X3 (~ 153). To the
cyclic subgroup C3 belongs the function
6 = (Xl - X2) (X2 - X3) (X3 - X1)
By Theorem 2 of ~ 154, a is a root of a quadratic equation
with coefficients in R. In fact, the discriminant of the cubic
equation is
2 = C12C22 + 18C1C2C - 4C23 - 4C133 - 27C32.