OF THE HIGHER DEGREES, WITH APPLICATIONS. 111 
can be formed ; where ¢ is an m** root of unity, and ¢ is a whole 
number less than m but not zero, and p involves only surds subordi- 
1 1 
nate (see §3) to 4 i or A 
By §47, one of the terms in (42) is equalto 7;. For our argument 
it is immaterial which be selected. Let 7, = 71. Therefore 
m—1 m — 2 1 
—(m 4," tad,” +....44, )=0. (47) 
The coefficients of the different powers of 4 here are not all zero, 
for the coefficient of the first power is ae Therefore by §5, an 
1 
equation¢d) = 1, subsists, ¢ being an m2 yoot of unity, and J; in- 
wig 
volving only surds exclusive of 4, ” that occur in (47). By Prop. 
1 
XIV., a is a surd of a higher rank (see §3) than any surd in (47) 
1 
except ri . Therefore we may put 
at a m—1 
h=dtd4” +d,4, +..-. +4n—14, nie 
~ 1 = 
where d, dj , etc., involve only surds lower in rank than 7 i eben 
Jie 
4, =" = (d + d; 4,” + ete.)™ 
1 2 
t , — , 
=—=d+d,4, + & 4, + ete; 
; ots 
where d , dj, etc., involve only surds lower in rank than Ai y By 
aly 
§8, since a is a surd in the simplified expressions 7; , the coefficients 
dl — d,, a , ete., in the equation 
8 
