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

FUNCTION DOMAINS, EQUALITY

293

through a point of A(cl,..., c,)=0. The equation A(a,,..., an-, v)
=0 in v has only a finite number of roots. Hence v can be varied continuously from an to An so that A (a,,..., an-i, v) O at each intermediate
point. The combination of our two paths gives a continuous path from
(a) to (ai,..., an-i, An) not passing through a point of A=0.
Similarly, there exist constants fi for which
(,,,...,-2, a, n-,2,, A... -, An-, An)O.
By hypothesis, there is a continuous path frcm (al,..., an-i, An) to
(l1,.. 2, an-2 -i, An), not passing through a point of A(cl,...
on-2, an-1, An)=0 and composed of points with Cn-l=an-i, Cn=An, and
hence not passing through a point of A(cl,..., cn)=0. Evidently
there is a continuous path from our final point to E=(f1,..., 3n-2,
An-1, An) not passing through a point of A==0. We now have a continuous path from (a) to E not passing through a point of A= 0. Proceeding
in this manner, we finally get such a path from (a) to (A).
Let xi0,..., xn~ be the n distinct roots of the equation with the
coefficients a,..., an. These roots receive increments as small in
absolute value as we please when ai,..., an are given increments sufficiently small in absolute value.* Hence if we proceed along our path
from (a) to (A), we obtain a definite coordination of the roots x'i,..., x'n
of the equation having the coefficients A,,..., An with the initial values
x~,..., xn~. Thus the latter and definite paths radiating from (a)
lead to n functions x1,..., xn of ci,..., c, uniquely defined for every
set of c's for which A/#0, and called the roots of the general equation (1).
In fact, for a particular set of c's, the roots of the equation are the values
of the functions xi,..., Xn for those c's. Our main investigation is the
comparison of a rational function of the roots with that derived by a
substitution on the roots; hence we shall not be interested in values of the
c's for which A==0, i.e., for which two or more roots become equal.
The same scheme defines a fortiori the roots of any equation whose
coefficients are functions of one or more variables. We retain only those
sets (A) which are sets of values of our present coefficients. The fact
that certain of the sets intermediate to (a) and (A) are not now values of
our coefficients does not disturb the coordination of the roots at (A) with
those at (a). The scheme therefore assembles the root values into root
functions.
152. Function Domains, Equality, Group of an Equation.
Instead of a domain composed of constants, we now employ
a domain R(k1,..., k,) composed of all rational functions
*The roots are continuous functions of the coefficients. For a proof, see
Weber's Algebra, vol. 1, 1895, p. 132.