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

~ 153]

GROUP OF THE GENERAL EQUATION

295

other than identity, on vi,..., v,. For every set of v's,
one factor of P is a value of {(xl,..., x,) and hence is zero.
Thus P is zero identically in the v's. Hence some factor 4' of
it is zero identically.
THEOREM. The group of the general equation for the domain
R defined by its coefficients and any chosen constants is the symmetric group.
The coefficients of its Galoisian resolvent G(V)=0 are
rational integral functions of ci,..., cn with constant coefficients. Replace ci,..., c by the elementary symmetric
functions of xi,.., Xn. Then G(V) becomes a polynomial P(V) whose coefficients are rational integral functions
of the x's with constant coefficients. Let Vi=Zmixi, where
mi,..., mn are distinct integers, be the function used in
constructing G(V). Then P(V1) is a rational integral function of the x's with constant coefficients which is zero for every
set of values of the c's. By the Lemma, P(V1) is zero identically in the x's. The function derived from it by applying
any substitution s on the x's is therefore zero identically in
the x's. Since the coefficients of G(V) are unaltered by this
substitution, we get G(Vs)=0. Hence every substitution s
occurs in the group of the equation.
EXERCISES
1. Prove the last theorem by showing that properties A and B in ~ 149
hold when G is the symmetric group. Note that if = os for all values
of the c's, then, by the Lemma, 0-As identically in the x's; if this is true
for every substitution s, < is symmetric and hence is a rational function
with rational coefficients of cl,..., c, and the coefficients of p. Next,
if +(xi,..., xn) equals a rational function of the c's and hence a rational
symmetric function i of the x's, for every set of c's, then 0-= identically
in the x's (Lemma), so that 0 is symmetric, and property B holds.
2. If G is the group of f(x, c)x xn-cx "- 1+...cn =O for the domain
R=R(cl,.., c,, ki,..., k1), where ki,.., ki are constants, and
if c'I,.., c'n are values which ci,..., c. can take, then the group
G' of f(x, c')=0 for R'=R(c'l,..., c'n, k,..., ki) is Gor a subgroup
of G.
Hint: If G(V, c) =0 is the Galoisian resolvent of f(x, c) =0 for R, the
group of f(x, c') =0 for R' is G or a subgroup according as G(V, c') is irreducible or reducible in R'.