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

ABELIAN EQUATIONS

211

Ex. 1. Every sub-group of an Abelian group is itself an Abelian group.
Ex. 2. If G1 is not Abelian, and G1 is a sub-group of G, then G is not
Abelian.
Ex. 3. Show that G3(3), G2(4), G4(4) I, G4(4) II, 4(4) III, G5(), GC(5) II,
are Abelian groups.
186. Abelian Equations have Abelian Groups. If the roots of
an Abelian equcatio, ar e all distinct, its Galois group is an
Abelian grou1p.
Let f(x) = 0 be an Abelian equation, and let its roots be
aC, - = 1(a), a2 -= 2(a),., N, =-l i-,(a). I
If f(x) = 0 is reducible, let g(x) be an irreducible factor, and
let g(x) = 0 have the roots
c/, C'= '(a), ax" = "('(),... II
All the roots of II occur, of course, in the series I. Now
g(x) = 0 satisfies all the conditions of a Galois resolvent of
f(x) = 0, ~ 145. Hence the group of f(x) = 0 consists of the
substitutions
p - ("aa), p' (.c~'), (.'.
This group obeys the commutative law in multiplication, for
we have ( (a ))
p" =(ta") =(C,."(o()),
and, ~ 148, p'p" -a= '(a)I f'(ta), 4'") '(a) a '"()t
p'p = o, P '"(o()I It"(al),."d'(a) } = r,,."c'(a)O.
Since the equation f(x) = is Abelian, we have
"'4Y'(a) =: '."(~);
hence, p'p = p"p.
Consequently, the group of substitutions of the domain Q()
is commutative, as is also the isomorphic group of the equation
f(x) = 0, ~ 151. Therefore, the Galois group of f(x) = 0 is an
Abelian group.