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

128 THEORY OF EQUATIONS
that it has then six distinct conjugate values; that when k = - 1 or k =,
the conjugate values are forially different; that the numerical values coincide in pairs when k -, and in triplets when k a, w being a complex
cube root of - 1. See ~ 111.
Ex. 11. Find the values of the roots of x - x3 - x+ -- 0, and show
that, for these values, the function ca2cx + (xitz2 +- (tZa2o2 + cIC32 does not
belong to the cyclic group, although this function is formally altered by
all substitutions in G24(4) which do not occur in G4T4)I.
* Ex. 12. Show that, for the general quartic, the following functions
belong to the cyclic group:
(a + 2 al) ((oc + 2 a2)(C2 + 2 (3)(aC3 + 2 a),
al(cc(a2 + 2 (al) + a13a2(cc12 - 2 c2) + 2a3(Ua22 + 2 a3) + a33a (a32 + 2 (t).