82 PRINCIPLES OF THE SOLUTION OF EQUATIONS 
§5. If an algebrical expression rj, arranged as in (1), be zero, while 
the coefficients g1, /), etc., are not all zero, an equation 
1 
™ 
od = (2) 
must subsist ; where w is an m root of unity ; and 4 is an expression. 
1 
involving only such surds exclusive of 4” as occur in 7;. For, let 
1 
the first of the coefficients hj, e3, etc., proceeding in the order of the 
1 
descending powers of 4™ , that is not zero, be 7, the coefficient of 
1 
§ 
4”™. Then we may put 
1 
1 s 
m 
mr, = mi f ( 4 yt = Ny a” + ete. = @ 
1 
™m .«. . ” 
Because 4 is a root of each of the equations f(x) = 0 and 
am — 4, = 0,f (a) and 2™ — 4; have a common measure. Let 
their H. ©. M., involving only such surds as occur in f(a) and 
a” — di, be g (x). ‘Then, because ¢ (x) is a measure of a — 4), 
the roots of the equation 
G(x) = 2 + pyres 4) pore =" + ete? = 
1 1 1 1 
m m mm . 
ared, , a4, , wed)! ,....,%¢—1 4, 5; where w, w2, ete., are dis- 
tinct primitive m* roots of unity. Therefore, 
A,” (wr o2..) (— 1)? = pe 
Now c is a whole number less than m but not zero; and, by §1, m is. 
prime. Therefore there are whole numbers n and / such that 
cn 1 
om mh ae 
4,” (wr 02. .)"(— 1) = 4,” 4, (ws o2..)"(— 1)™ = p.. 
if 
h n 
Therefore, if (w, w2..)” = w, and J; 4, (— 1) = Py Ay = h. 
§6. Let 7; be an algebraical expression in which no root of unity 
; 1 
having a rational value occurs in the surd form 1™. Also let there 
1 
. ™m . 
be in 7; no surd 4 not a root of unity, such that 
