162 RESOLUTION OF ALGEBRAICAL EQUATIONSe 
and the surds U and Y are similarly involved in f(p): which is not 
the case. Hence f (») contains no surds, except Y and its subordi- 
nates. Let T be a chief subordinate of Y, with the index S, Then, 
since the coefficients of the different powers of Z in X are rational, 
T disappears from X. Let the factors by the continued product of 
which X is produced be [compare (1) Prop. XI] 
274 
{a—f (p)} ov F(a), Fe (@)s «..-.- poke (thos AG) 
and, when T is changed into 2,T, z, being a o root of unity, dis- 
tinct from unity, let the terms in (3) be transformed into 
Xe COON eee a TOY Or. Oia 
Then, since the terms in (4) are the five factors of X, F, (w) must be 
equal to one of these terms, which (on the principle pointed out in 
Prop XI.) may be assumed to be ,X,. Consequently (Prop. XI.) 
A 
SAT NOMS (oY ER UGR MR REE EN Me Oni RoHS ibe (13) 
where Y, is what Y becomes when T is changed into 2,T; and P is 
an expression involving only surds in f (7), distmct from Y; 2 being 
a whole number, neither zero nor unity, less than 5, and such that 
Ne Shee ee ce ae On een 
where wis a whole number. Since o is a prime number, the only 
values of A, less than 5, which satisfy equation (6), are land 4 
And X is not unity. Therefore \—4. Hence o—2; and T is of 
the form, T= ./C. Next, suppose, if possible, that U is a chief 
subordinate of Y, distinct from T. By the same process of reasoning 
as above, it may be shown that the index of U is }; and, if ,¥ be 
what Y becomes when U is taken with the negative sign, —U, the 
equation, 
Y = Q (x*) 5) 
subsists ; where Q is an expression such as P in (5). Therefore 
4 
Te WaT Oa . 
By raising both sides of this equation to the fourth power, keeping 
in view that the common index of Y, and ,Y is, we get 
y Nic Vis sieve idinse ape ne. yee vance me ea ee) 
where Ris an expression, like P and Q, clear of the surds Y, and ,Y. 
