OF THE HIGHER DEGREES, WITH APPLICATIONS. 97 
the series of integers m, c, etc., of which JV is the continued product, 
is reduced to its first term. If 7; involve only surds that are roots of 
unity, 2 — 1 is a multiple of V; for VY=m.... 4g; therefore, 
because WV is prime, it is equal to m; but ms = n — 1; therefore 
n—1= sWN. 
THE SoLvaBLe [RREDUCIBLE EQuaTION OF THE mh DEGREE, m PRIME. 
§30. The prixciples that have been established may be illustrated 
by an examination of the solvable irreducible rational equation of the 
m‘h degree F(x) = 0,m being prime. Two cases may be distinguished, 
though it will be found that the roots can in the two cases be brought 
under a common form ; the one case being that in which the simplified 
root r; is, and the other that in which it is not, a rational function of 
roots of unity, that is, according to §1, of roots of unity having the 
denominators of their indices prime numbers. The equation /' (a) = 0 
may be said to be in the former case of the first class, and in the latter 
of the second class. 
Tue Equation /' (x) = 0 or THE First Cuass. 
§31. In this case, by Cor. Prop. VI., 7; being modified according to 
§21, if one of the roots involved in 1; be the primitive n root of 
unity w;, ~ — 1 is a multiple of m. Also, the expression written 
X, in Prop. VI. is reduced to x — 71, so that 
ry = M10; + pele + .... + DmCm. 
The m roots of the equation F (x) = 0 being 71, 72, etc, we must 
have 
= 71 Cy t+ polo + .... + pmCn, 
72 = PmC1 + prC2 +e... +H Pini mes (23) 
m= p2Cy + psCo + .... + mCm. 
For, by Prop. II., because r; is a root of the equation F (x) = 0, all 
the expressions on the right of the equations (23) are roots of that 
equation. And no two of these expressions are equal to one another. 
For, take the first two. If these were equal, we should have 
(pm — Pr) Ci + (pi — p2) Co + etc. = 0. Therefore, by §13, 
each of the terms pm — 1, ~1 — pz, etc, is zero. This makes 
Pi, p2, ete., all equal to one another. Therefore 7, = — py ; so 
that the primitive n** root of unity is eliminated from 7; ; which, by 
§21, is impossible. Hence the values of the m roots of the equation 
F (x) = 0 are those given in (23). 
