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

186

THEORY OF EQUATIONS

Since G4(4)III is an intransitive group, the quartic can be
factored in the domain ~'(?,,T,. The two quadratic equations
thereby obtained have as Galois groups 1, (ab), and 1, (ccl),
respectively. From VI, ~ 62, we see that 2'(~s;, v= v '(vw, v,).
Hence it is not necessary to adjoin more than one of the two
irrationals \/u, \/w.
The quartic offers a better exhibit of the Galois theory than
did the quadratic and cubic equations, because not only may
we select a great variety of different functions Mi at each
adjunction, but we may select different groups. In the above
solution the series of groups taken is G24,,, G,4), G4(4)III,
G = (1, (ab)), G = 1, but another series may be chosen, viz.
G24(4), G12(4), G4(4)II, G(4), 1. In Exs. 1 and 3, ~ 163, a solution
of the quartic is outlined, in which this series of groups is used.
Again, we may effect a solution by first adjoining a function
that belongs to the cyclic group G4(4)I; say,
y = aci1 4 2 2 - 2- + (t2 + a(t,2.
To be sure, the first resolvent equation q (y) = 0 will be of
the sixth degree, but it can be treated as an equation of the
third degree and a quadratic.
The number of different solutions of cubic and quartic equations which have been given since the time of Tartaglia and Cardan is enormous. For information on different solutions consult
L. Matthiessen, Grundzilge der Antiken u. Modernen Algebra.
It would seem that the above mode of procedure should lead
to solutions of the general quintic equation. But an unexpected
difficulty arises in our inability to solve all the resolvent equations. There arise resolvents of higher than the fourth degree.
The Galois theory will furnish proof that the solution by radicals of the general quintic and of general equations of higher
degrees is not possible. In the remaining chapters we shall
demonstrate this impossibility and discuss the theory of special
types of equations of higher degree which can be solved
algebraically.