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

216

THEORY OF EQUATIONS

is a function of x in Q ), ~ 154. Since all the roots of
f(x, M) = 0 are roots of f(x) 0=, f(x, M1) is a factor of f(x)
in Q,^).
Similarly, we can show that
f(z, M) =- (X - ). (X - ), -
f(x, 2) - (x - y) (x - yl).. (x - yr-1), etc.,
are factors of f(x). We have, therefore,
f(x) -f(x, 31). f(x,:11i) A... f( i ).
Since the coefficients of f(x, 3I) = 0 are cyclic functions of
its roots, the group of this equation is the cyclic group, or one
of its sub-groups, ~ 159. But a cyclic group can have no transitive sub-group, hence the irreducible equation f(x, 11) = 0 is
a cyclic equation. Similarly for f(x, M1) = 0, etc.
It remains to explain how the values of MV, -.., 1/-i may be
obtained. By ~ 161 they are roots of an irreducible equation
g(M) 0 in 0 of the degree t. We proceed to prove that
g(M) =0 is Abelian. Since f(x, M) =0 is cyclic, we get for
the conjugates of M,
M -= [C, (c), *.., fr- (c)] = F(a)
Mx=, [,, ^(,)* r. * _1+l (,B)] ^) = (
M2_= [y, (y)/..., <,-1 (y)] = F()............. j
By assumption, we have / = c((a), y,= 1(t). Hence
M= _ [I(c), 4~(),, *4 ")-',(a)]
= ~[,(c),,,(,),..., ~~"-1'()]
= 1i:,[4,., *,^_],
where i, admits the substitutions of the cyclic group. Hence,
by ~ 162, M3 is a function in Q of 3M. Similarly for Mi.