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

224

THEORY OF EQUATIONS

whose index is a prime number j. Continuing in this way, we
finally arrive at the identical group 1. This proves the
theorem.
197. Criterion Applied. The Galois group of the general
equation of the nth degree is the symmetrical group of the nth
degree. The symmetric group has always the alternating
group as a sub-group. This alternating sub-group is a normal
sub-group of the index 2. It becomes the group of the given
equation by the adjunction of the square root of the discriminant. The principal series of composition, ~ 110, is G,'" 1, for
the quadratic; G(3), G(3, 1, for the general cubic; and G.4),
G,4), G(4) II,,(4, 1, for the general quartic. In these cases the
alternating group is seen to have a normal sub-group of prime
index. We are going to show that when the degree of the
general equation is greater than 4, and, consequently, the
degree of the Galois group is greater than 4, the alternatilng
group has no normal sub-group of prime index.
198. Theorem. An1 alternating gr'oup of higher degree than
the fourth has no normal sub-group of prime index.
All substitutions of an alternating group are even, ~~ 99,100,
and are expressible as the product of cycles of three elements,
~ 93. Let these substitutions be so expressed.
We first establish the possibility of selecting a substitution
s in the alternating group, so that a given cycle of three
elements, say (1 2 3), will be transformed into any other cycle
of three elements occurring in the alternating group. Suppose
that 1, 2, 3, 4, r, t, u, v, are elements of the group and we wish
to transform (1 2 3) into (r t u). It is easily seen that the
substitutions =(12 3 4) will do it; for, s-1 (12 3)s = (r t i).
r t it \
That s is a substitution in the alternating group is clear, since,
~ 82, s = (1 2 t) (1 2 r) (3 4 v) (3 4 u), an event substitution.