OF THE HIGHER DEGREES, WITH APPLICATIONS. 103 
‘Therefore, by Prop. III., Dp is the root of a simple equation, or has a 
rational value. In like manner each of the expressions 
Soh Beh el i. | ees (33) 
has a rational value. From (26), by actual multiplication, 
ote! 
—_ 
1 (ihe 
Ay 4", = Sr + (32) + (23) GF ete. 
But = = , etc., are respectively identical with Sis Se 1, etc. 
Therefore 
1 i 
™m m 
A A = t+ (R) (A + A) + (Sali + AH) + ete.(34) 
Hence, since the terms in (33) are all rational, and since the terms in 
1 1 
. nm mm : 
(32) are respectively what 4, 4, _ , becomes by changing f succes- 
: . —1 , 
sively into the — terms t,, tf], etc., the terms in (32) are the 
m — 1\th 
2 
roots of a rational equation of the ( degree. 
§40. For the solution of the equation a" — 1 = 0, n being a prime 
number such that m is a prime measure of m — 1, it is necessary to 
obtain the solution of the equation of the m‘* degree which has for 
one of its roots the sum of the terms in a cycle of primitive 
m4 roots of unity. This latter equation will be referred to as the 
reducing Gaussian equation of the m** degree to the equation 
ge —1=0. 
§41. Proposition XI. When the equation / (x) = 0 is the re- 
ducing Gaussian (see §40) of the mt degree to the equation 
a” — 1 = 0, each of the —— 
expressions in (32) is equal to n. 
Let the sum of the primitive nt” roots of unity forming the cycle 
(8), which sum has in preceding sections been indicated by the 
symbol Cj , be the root 7; of the equation / («) = 0. This implies, 
since s is the number of the terms in (8), that ms =n — 1. Let 
us reason first on the assumption that the cycle (8) is made up of 
. . —1 
pairs of reciprocal roots w; and w; , and so on. Then, because the 
: ay : eae ae 
cycle consists of - pairs of reciprocal roots, Cy or vr; is the sum of 
ad 
