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

CHAPTER XVIII

ABELIAN EQUATIONS
184. Definition. An equation f(x)= 0 of the nth degree,
having the roots a, a1,., (_n- is called Abelian, if each root
can be expressed as a function in 12 of some one of its roots,
thus,? = l bl((), (2- = 2 2(a), *. *,,_1 =,_-l(a),
and if, for any two of these roots, we have the commutative
relation hk() = kh(a). I
By 0hk((a) we mean here,h[0k(a)].
The equation x41 - 0 is Abelian, because, its roots being
~ 1, ~ i, we have -1 = i2, - i3, 1 = i4, (i2)3 = (i3)2, etc.
Ex. 1. Show that cyclic equations are special cases of Abelian equations,
Ex. 2. Show that xG - 1 = 0 is Abelian, but not cyclic; that x3 - 1 = 0
is both Abelian and cyclic.
Ex. 3. Prove that when Abelian equations are irreducible, they are
normal.
Ex. 4. Show that xn - 1 = 0 is Abelian where n is any positive integer.
Ex. 5. The equation x5 + 22 x -440x3 - 3520 x + 11264x + 32768 = 0
has as three of its roots - 2, 4, - 8. Show that it is an Abelian equation.
Ex. 6. Is x6 - 5 = an Abelian equation in the domain 2(I)? In the
domain l(i,.), where w is a primitive sixth root of unity?
185. Abelian Groups. A group whose substitutions obey the
commutative law in multiplication is called an Abelian group.
For instance, 1, (ab) is such a group, because 1. (ab) = (ab) 1.
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