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

212

THEORY OF EQUATIONS

187. An Equation having an Abelian Group is Abelian. An
irreducible equation g(x) = 0, having a commutative group is an
4belian equation.
Let a, acl.., a,,_x be the roots of g(x)= 0 and let G represent
the group of this equation. As g(x) = 0 is irreducible, G is
transitive, ~ 156.
Let s be any substitution in the group G which does not
change the digit 0, and let si be any substitution in G which
replaces 0 by i. Then s-1 s. si is a substitution of G which
does not change i; for
Si-l changes i to 0,
s does not change 0,
si changes 0 to i.
Since the group G is assumed to be commutative, we have
Si-1. ' Si == si-1' Si ' s = s.
Hence s leaves unchanged not only the digit 0, but also the
digit i. But the group G is transitive; therefore, the digit 0
must be capable of being replaced by each of the other digits
1, 2, 3, ***, (at-1). Yet, no matter which one of these digits
is taken to be i, the substitution s leaves i unaltered. These
relations can hold true only when s is the identical substitution
in the group G. Hence every substitution in G, except 1, replaces 0 by some other digit.
Applying to every other digit the same reasoning which we
applied to 0, it follows that every substitution in the group G,
except the substitution 1, contains that digit among its elements;
in other words, there is no substitution in G, except 1, which
leaves any digit unaltered.
Next, adjoin to the domain Q the quantity M-= a, where a
is one of the roots of g(x) = 0. Since no substitution in the
group G, except 1, leaves the index of a, unaltered and since
the identical substitution satisfies the definition of a group, 1 is
the sub-group to which M belongs. Thus, Q= 1; and, by the