142 RESOLUTION OF ALGEBRAICAL EQUATIONS. 
Ss 2 2 Cul Co | 
@¢=D+4+D)D,2T+ D.%4T + .... + Do 2 ery 
where 2} is an indefinite ot root of unity; and D, D,, &c., are clear 
of the surd T. Then 
A 
AX peo To DT a ke, 
A 2 
id (BU) 24D. 214 Dys To eRe 
Ded = ea Ws Ce, ick ees ee 
This equation involves only such surds as occur in f (p). Therefore 
(Cor. 1, Def. 9) the coefficients, D; (1 — z), Dz (1 — 2), &c., vanish 
separately. But, since o is a prime number, and z is a o*} root of 
unity, distinct from unity, none of the terms, 1 — z, 1 — 2, &e., 
vanish. Therefore D,, Dz, &c., must all vanish; and 
1 
A Ss 
AY =D 
Raise both sides of this equation to the 7th power; r and x being 
whole numbers such that 
rrX=ns+ 1. 
1 
r n 3 
Then, (AY) =(A Y )Y=(D')’, 
1 
iPM Oye a kt eee (3) 
where P and Q involve only such surds, exclusive of Y, as are present 
in A va and Q is clear of the surd T. Let the forms of P and Y be, 
2 1 
Peete Bad abe Dae Lc eee Th le alee (4) 
I 
9 = — 
VO eh ee es ee (5) 
where 6, b,, &c., h, hy, &c., are clear of the surd T. Suppose,if 
possible, that the terms 0, , 4,, &c., are all zero. Then P= 0; and, 
s s 8 2 
bh+bBmMT+b kT + & =Q. 
But since Q is clear of the surd T, the coefficients, b Ay; p ho » & ee, 
in this equation, must (Cor. 1, Def. 9) vanish separately. Now, A is 
(by hypothesis) not zero; therefore P is not zero; therefore b is not 
zero, Therefore all the terms, 2y, 2, &c., vanish: which (since T 
