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

188

THE()-tY OF LQUAT1IONS

group is either the cyclic group of the degree n - 1 or one of
its sub-groups, ~ 162. Since I is a normal equation, it is its
own Galois resolvent; the Galois domain is of the degree n - 1
and the Galois group of the order n - 1. Hence the Galois
group of I is the cyclic group of the (n - l)th order.
Ex. 1. If n is prime, show that x' - 1 = 0 is a cyclic equation in the
domain 0(1). In what follows we shall exclude from our consideration
cyclic equations whose roots are not all irrational.
171. Theorem. Each root of a cyclic equation ca, be expressed
as c function i 0 of any other root.
If a, act, *, (a,,-_ are the roots of the cyclic equation f(x) = 0,
then the function in Q of x of the (n - 1)th degree,
b () a - f (X) C- +. *. + _ ----
X -- C1 X -- gtadmits the permutations of the cyclic group and is, therefore, a number in 2, ~ 154. If we put in succession
x = a,, *.., t,1_1, and if we use the notation (x) =- (X), we get,
f'(x)
~ 142, ac = (a), a2 = ~(a1), *, a,,_ = (c,, a_), =b(,,This holds even when f(x) = 0 is a reducible equation, provided that it has no multiple roots.
Ex. 1. When are cyclic equations normal?
Ex. 2. Show that one root of a quadratic equation can be expressed
as a function in (,(t,,t,) of the other root.
Ex. 3. Show that any root of a cubic can be expressed as a function
in Q(,a~, a2, a3, V) of one of the others.
Ex. 4. Show that a(2 = 2(a(c), act = 3(a), etc., where the superscript
is not an exponent, but indicates that the functional operation 0 is to be
repeated. Thus, 0'((/) = (q(c)).
Ex. 5. Prove that tli = npt+l(c(), c = Q"+2(a), etc.