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

200

THEORY OF EQUATIONS

Ex. 1. By trial find the smallest integer that may be taken as the value
for g when n = 11, and show that c, Wc, wg2,..: Cg1( represent the same
roots as wA, d w23, 3, w10. Show that, for n = 13, g may be 2 or 6.
180. Solution of Cyclotomic Equations reduced to Equations of
Prime Degree. As is evident from ~ 174 we can base the solution of equation I of ~ 178 upon cyclic equations whose degrees
are prime factors of in-i. When n is prime, n -1 is composite.
Let n - 1 = e.f, where e is a prime factor. As before, let o be a
root of the cyclotomic equation I. Then construct expressions
A, e... *e-i) called periods, as follows:
7 _ 0 + _ge + g2e + * + (}(f-l)e,
_ -g + @(_ge+l + o ~ge+1 + + og(/-l)e+l
= 'o~ ~ III
ge -1 + 92e-i + g3e-l + + gfe-l
In each period there are f terms and the first term is the geth
power of the last term, and each of the terms after the first is
the geth power of the term preceding it. Each of the periods
is, therefore, a function that belongs to the cyclic group
GT!=~~) Qe) S~e) ~.. S(f —)e
G= { 1, se s2e., }(,-l)e
where the substitution s= (o, I, W1,, w,2 * _. The periods III
are special forms which the functions y, yi, Y2 in ~ 174 may
assume. From ~ 174 it follows that the periods III are the
roots of an irreducible cyclic equation
($- 7) (X - l)... (.- e-1) = 0. IV
This is an equation in l and of the degree e. By the solution
of this equation the periods become known quantities.
181. Product of Two Periods. In order to compute the coefficients of equation IV in ~ 180 we must multiply periods
one by another. Take
gh - gre + +.. +(f-l)e
k - (g + - +' " + + ( -- )