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

192

THEORY OF EQUATIONS

Ex. 2. Carry out the computation in Ex. 1 by taking
al = a2 0, ag - a4
and show that T will have the values 60 j 80i, which lie in the
domain Q(1, i).
Ex. 3. Find TL and T2 when in the quartic al =a2 = a4 = 0, a = 1.
In this case, is the cyclic group the Galois group in 9(i,i)?
Ex. 4. Taking c - ae + a2 - a3 = Vp/,
cC + a1 - 2 - a3 = Vp2,
(a - Cal - z2 + o3 = / p,
give a solution of the general quartic, pi, p2, p3, being roots of
p3 + (8 a2 - 3 al2)p2 + (3 a14 - 16 a12a2 + 16 ala3 + 16 a22 - 64 a4)p
- (ai3 - 4 a1a2 + 8 a3)2 = 0. See Ex. 1.
Ex. 5. Find a solution of the general quartic by taking
a + ial - a2 - ia3 = T,
a- a 1+ a2- a3 =A-(/Ti)2,
a - ial - ao2 + ia3 = B( /T1'1)3,
where A = (- al - - a -3) (a + ia - a2 - iae3)-2.pi[ Ti + (al2 -2 -2- 2 p1)2]
2 T1(4 aa2 - a13 - 8 a3)
B = (a - ial - a2 + ia3) (a + ial - a2 - ia3)-8
a12 - 2 a20 - 2 0
T1
174. Cyclic Equations of Prime Degree. The solution of tany
cyclic equation can be made to depend upon the solution of cyclic
equations whose degrees are prime.
The solution in ~ 172 applies to cyclic equations of any degree
and is perfectly general. Nevertheless it is of importance, for
subsequent developments, to prove the present theorem. We
give the proof for the degree 12 = 3 * 4. The generalization to
the case n = e. f is obvious.