OF THE HIGHER DEGREES, WITH APPLICATIONS. 95 
§28. Proposition VI. Let the simplified expression 7, , modified 
according to §21, be a root of the rational irreducible equation of the 
1 
WV degree, / (x) = 0. Then if 4” , hot a root of unity, be a surd 
of the highest rank in7,;, V is a multiple of m. But if 7, involve 
only surds that are roots of unity, one of them being the primitive 
n* root of unity, V is a inultiple of a measure of n — 1. 
1 
First, let 4” , not a root of unity, be a surd of the highest rank 
in 7;. Taking the expression (1) to be 71, let Xy be formed as in 
§24, and let it be modified according to §21. It is clear of the 
1 
surd 4," . Should it involve a surd that is not a root of unity, let 
X» be formed as in §27. Setting out from 7, we arrived by one step 
1 
at X,, an expression clear of 4 ta , and such that the roots of the 
equation XY; = 0 are unequal particular cognate forms of R. A 
second step brought us to Xz, an expression clear of the additional 
1 
surd a. , and such that the me roots of the equation X2 = 0 are 
unequal particular cognate forms of #&. Thus we can go on till, in 
the series X,, X-2, etc., we reach a term YX, into which no surds 
enter that are not roots of unity, the mc .... / roots of the equation 
X, = 0 being unequal particular cognate forms of FR. Should YX, 
modified according to §21, not be rational, its form, by Prop. IV., 
putting d for mc .... J, is 
X,=ag—(piCit....+pmCm)a*-1+ (qiOit ....+¢mCm)xt-? + ete. ; 
where, one of the roots occurring in X, being the primitive n*® root 
of unity w, , the coeflicients p; , qi, etc., are clear of w, ; and C is 
the sum of the cycle of primitive m® roots of unity (8) containing 
wa | 
8 or terms ; and, the cycle (9) containing all the primitive 
nth roots of unity, the change of q@ ; into Ae causes (; to become C2, 
and C2 to become C3 , and so on, C,, becoming Cy. As was explained 
at the close of §20, the cycle (8) may be reduced to a single term, 
which is then identical with Cy. It will also not be forgotten that 
the roots of unity such as the n'* here spoken of are, according to §1, 
subject to the condition that the numbers such as x are prime. When 
C; in X, is changed successively into C1, Ce, etc., let XY, become 
4 
: ~(m —1) 
ie ave... ae (22) 
