An introduction to the modern theory of equations, by Florian Cajori.

126

THEORY OF EQUATIONS

If, however, we apply to the functions I a substitution of Gr
which does not occur in G0, we obtain a series of functions
P% P, '" P m-l
of which at least p' does not occur in I. For, if p' did occur
in I, we would have two identical functions, distinct from p,
resulting from the application to p of two different substitutions. This is impossible.
If now we form a new function P thus,
~ = (t - P) (t - pi)... (t -p-1)
where t is a variable, it is evident that p remains invariant
when operated on by the substitutions of the sub-group Gl, but
varies for any substitution in G which does not occur in G,.
Hence q is a function which belongs to G1, taken as a sub-group
of G.
We ar at liberty to assign to t any rational value which will
keep ~ distinct from any value obtained for it by application
to ~ of a substitution in G that is not in GI. One such value is
t=0.
113. This method of finding functions belonging to a group
does not usually furnish simple results directly, as will be seen
from the following example.
Ex. 1. Form a function of ai, c2, (a, a4, which belongs to
2(4) _ 1, (1 3) (2 4), taken as a sub-group of
G4(4)I 1, (1 3)(2 4), (1 2)(3 4), (1 4)(2 3).
Assume p = calt + c2(a, + C.3a -+- c44, such that p assumes four distinct
values for the substitutions of G4(4)II. The substitutions of G2(4) applied
to p yield
to p yield p cl1Cl 4- c2(t2 + c3(C3 + C4oC4,
Pi = clC3 + C2t4 + C3tl1 + c4ct2,
hence i = (t- p)(t- pi) =t2- (atl + 3) (tC1 + tC3) -(Ct2 + a4)(tc2 +tc4)
+ (Ct12 + a42)C1C3 + ( (22 + CL42) C2C4
- (ctlt3(C12 + C32) + Cg2g4(422 + C42)
+ (c2(23 -+ cctl4) (C1C2 + C3C4) + ((t1(t2 + (f34)(C1C4 + C2C3).