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

(YCLIC UEQUATIONS

193

Let s = (Kca..,), where a,= G3(c, = (a), a3 = (2), *'*"
then s3 can be resolved into three cycles, c, cD, c2, as follows:
C = (aaag6Ct)
c = (a1Ual4alo),
C2 = (a2a5at8ll).
Let y be a function q iln 0 of the roots a, aC, a, a,, which
belongs to the cycle c. The substitutions of the Galois group
P= s1, s, s2, *.. Sn-l of f(x) = 0, applied to y, give three distinct
values,
y = q(aa3a6o),
Y1 = (C14CCo10),
Y2 = A(U2t5U8cll))
which are roots of a cubic equation,
(t -y)(t - y1)(t -y2 =0. I
The coefficients of t in I are symmetric functions in Q0 of y,
yi, Y2, and are, therefore, unaltered by the substitutions of P.
Hence these coefficients are numbers in 2, ~ 154.
We proceed to show that I is a cyclic equation whose group
is P1 =' 1, (YY1Y2), (YY2Yi). Rnemebering that the substitutions
of the group P interchange y, y, Y2 cyclically, we see, firstly,
that any function of y, y1, Y2 which admits of the substitution
of P1 is a function of a, a,,..., ca_- which admits of the substitutions of P (the Galois group of f(x) = 0), and such a function
is a number in Q, ~ 154; secondly, any function of y, y', Y2,
which is a number in Q9, is a function of the roots t, a(,.**, a_,nwhich is a number in 0 and hence admits of the Galois group P,
~ 153, thus showing that the function of y, y1, y, admits of the
substitutions of P,. Consequently P1 is the Galois group of
equation I, ~ 155.
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