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

288

GROUP OF AN ALGEBRAIC EQUATION [CH. XIV

X(V).-ru(V) = 0 with coefficients in R is satisfied by every
root V, of the irreducible equation G(V) = 0 (~ 144). Hence
- (VS) 4S
so that q/4 is unaltered in value by the substitutions of G.
EXAMPLE. Consider a cubic equation, like (2), with a rational root
xl and no multiple root. By property B with xl as the rational function,
its group for any domain containing the coefficients has no substitution
other than 1 and (x2x3). If the domain contains x2 and hence also X3,
the group is the identity; this is the case with equation (2) for R(i). In
the contrary case, there must, by property A, be a substitution altering
x2, so that the group is {1, (x2x3)}.
Since an n!-valued function V1 with coefficients in a given
domain R can be chosen in an infinitude of ways, there are
infinitely many Galois resolvents G(V)= 0. Our definition
of the group G of the given equation for the domain was based
upon a single such resolvent, i.e., upon a particular V1. It
is a fundamental proposition that different functions VI always
lead to the same group G. This follows from the
THEOREM. The group of a given equation for a given
domain R is uniquely defined by properties A and B.
First, suppose that G'={1, a', b',..., m'} is a group
for which property A holds. Then the coefficients of
(v)-(v -V1)(V-Va,)(V-Vb,)... (v-Tn),
being symmetric functions of V,..., V,, are unaltered
numerically by the substitutions of G' and hence equal numbers in R. Since the equation q(V)=Q, with coefficients in
R, admits one root Vi of the irreducible Galoisian resolvent
G(V) =0, it admits all of the roots (6) of the latter (~ 144).
Hence 1, a,..., I occur among the substitutions of G', so
that G is a subgroup * of G'.
Second, suppose that r= 1, a, 3,..., x} is a group
for which property B holds. Then the Galoisian function
* In Part III, a group is included among its subgroups.