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

~ 156]

ADJUNCTION TO THE DOMAIN

299

rational function 4 of xi,...,,,, then O equals a rational
function of A4 and the elementary symmetric functions
C1=-X1,..., Cn=XlX2... Xn.
The group of the equation with the coefficients ci,...,,
for the domain R, defined by ci,...., cn and given constants
ki, is the symmetric group. For, property A then states that
any symmetric function -of xl,..., x, with coefficients
rational in the k's is in R, and is the well-known theorem on
symmetric functions. Conversely, a function equal to a quantity in R is symmetric.
EXAMPLE. y2=XIX3+X2X4 is unaltered by all of the substitutions 1,
(13), (24), (13)(24), which leave I=x1+X3-X2-X4 unaltered. We see
that
y2,=i4(2-c12+4C2) C1=X, c2==1, 2= X2.
When a rational function of independent variables x,...
x, is unaltered by each substitution of a group H on the x's,
but is altered by every substitution not in H, it is said to belong
to the group H. We need not specify as in ~ 154 that H is a
subgroup of the group G of the equation with the x's as roots,
since G is now the total symmetric group..
156. Effect on the Group by an Adjunction to the Domain.
Let G be the group of f(x) =0 for a domain R =R(ki,..., kmn)
containing the coefficients. Let R'=R(4, kl,..., ki) be
the domain composed of the rational functions of 4, k,..
km with rational coefficients. This enlarged domain R' is said
to be derived from R by adjoining the quantity 4. If the
irreducible Galoisian resolvent G(V)=0 for the initial domain
R remains irreducible in R', the group of f(x)= 0 for R' is
evidently G. But if it reduces in R', let G'(V) be that factor
of G(V) which has its coefficients in R', is irreducible in R', and
vanishes for V==V1. Then if V1, Va,..., V, are the roots
of G'(V) =0, the group of f(x) =0 for R' is G'= {1, a,..., k},
a subgroup of G. As a group is included among its subgroups,
we have
THEOREM 1. By an adjunction to the domain, the group
of an equation is reduced to a subgroup.