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

CHAPTER XVII

CYCLIC EQUATIONS
170. Definition. A cyclic equation is one whose Galois group
is the cyclic group, ~ 101. Kronecker called such equations
"einfache Abel'sche Gleichulmgen."
A quadratic equatiorn is, cyclic; for the Galois group is the
symmetric group G2(, which is at the same time the cyclic
group of the second degree.
The general cubic is not a cyclic equation in the domain defined
by its coefficients; for its Galois group is G(), which is not a
cyclic group. However, if we adjoin
VD =- (a - R1)2a 1- f2)(a1 - a2),
the Galois group becomes (~ 163) G(3, which is cyclic. Hence
the general cubic is cyclic in the domain ~(a,a, a,,VD))
The general qtuartic is not a cyclic equation in the domain
defined by its coefficients, but if we adjoin a function which
belongs to the cyclic group G4~)I, the equation is cyclic in the
new domain. One such function that may be adjoined is
M = 3ac12 + (1(92 + t2 + a a2.
If n is a prime number,
xn-l + xn-2 +... + aX + 1 = I
is a cyclic equation in the domain Q(). For, ~ 130, this equation is irreducible. The cyclic function
012W2 + 2 + s. + W. +,^_-2W
is seen by the relations o2 - o12, 03= o13, etc., to be equal to
the sum of the roots, which is - 1. Therefore the Galois
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