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

SYMMETRIC FUNCTIONS OF THE ROOTS

91

We have al = 0, and therefore a2 = Zppl - - p2 - 6. Multiplying
Eppl by Mp, we have 3 Zppp2 + Zppl'2 -= 0, hence
a3 = - ZPP1P2 = 1 Zpp2 - -- - P3 p - 10.
Multiplying ZPP1P2 by Zp, we obtain
4 Zppip2p3 + Zppip22 = 0; Ippip22 = Sp22 * pp1 - Zp23 ~* p + Zp24
= Zp22Z'pp + Zp24 =- 48,
hence a4 = 12.
Similarly, we get
5 Zpp1P2p3p4 + MPP1P2P32 = 0, Ppplp2p32 = Zp32 ~* pp1P2 -p33 * Zppi
- p35 =- 300,
hence a5 = - 60. We have a = 17.
* Ex. 15. Find the value, in terms of the coefficients of the cubic, of
(c + wai + w2a2)3 + (a + w2a1I + wCa2)3, where c is a complex cube root
of unity.
* Ex. 16. Show that for the quartic
X4 + 4 blx3 + 6 b2X2 + 4 b3g + b4 = 0,
the following relations hold:
Zac5ax = 1536 b14b2 - 2304 b12b22 + 432 b23 - 256 bl3b3 + 672 blb2b3
- 48 b32 + 16 b12b4 - 36 b2b4.
Zca4(1a2 = 256 bl3b3 - 288 bib2b3 + 48 b32 - 16 b12b4 + 12 b2b4.
2 ta3Ca12X = 96 blb2b3 - 48 b32 - 48 b12b4 + 24 bab4.
Zca3i3 = 216 b23 - 288 blb2b3 + 48 b32 + 48 b12b4 - 18 b2b4.
a2a2X12(X2a3 = 6 b2b4.
Ct12 = 16 b2 - 12 b2.
IZaalca2 = 16 bib3 - 4 b4.
* Ex. 17. Find the cubic whose roots are
(a - al)(x2 - a3), (a - at2) (a3 - (1), ((t - 3) ((l - a2).
* Ex. 18. Show that, for the quartic x4 + a1x3 + Ca2X2 + a3x + C4 = 0,
we have
(a + ai - a2 - aa)(a - C - a2 + a3)(a- al + a2 - a3)
- (a13 - 4 ala2 + 8 a3).