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

CYCLIC EQUATIONS

201

Observing that 1h remains unaltered when o(h is replaced by
og^he or by any of the other roots in that period, we may write
the product of the two periods as follows:
IfG + h+ +..e gh+(/-I)e
+ gk+e (09 +e + a +' e + + gh+fe)
+...........,.
+ gk+(f-l)e( 97+(f-1)e + h +.fe +.. + +2(f-l)e)
In this product the terms in the first column are,
o(gkgh) + W(gk+gh)ge + (gk gh)g2e... _+ (gk+gh)g(f-l)e
If (gk +gh) is a multiple of n., then this column becomes
equal to. If (C/' + '1) is not a multiple of n, then this column
is one of the periods in III, ~ 180.
The same conclusion is reached for every column in the
product. Hence the product is a linear function of the periods,
the coefficients in this function being numbers in the given
domain Q (1).
182. When f is a Composite Number. When in the relation
n- = e f, both e and f are prilme numbers, the solution
of the cyclotomic equation is evidently made to depend on the
solution of two equations whose degrees are prime, one equation being of the degree e, the other of the degree f.
When f is a composite number, one or umore additional steps
are necessary to reduce the problem to the solution of equations of prime degree. If f= e'.f', where e' is prime, we may
form ee' periods, with f' terms in each, as follows:
I _ ccee' 2ee' (f'-l)ee'
'1 - (9 + 9 +, ~ ' + ~ (+ f1)
I _ Ie e+ e'C+e e+e 2'+e ('-l)ee'+e
l72e DW92e g ee'+2e + 2ee'+2e (/'2-l)ee'+2e
7' 2e + + + '+,). q_ ogee+ee'-1 2e'-1 3 e e'- ( 'g'-1
e) cc'-1, Dge' +.n + +.. + g/ +4