84 PRINCIPLES OF THE SOLUTION OF EQUATIONS 
third root of unity not being counted. If 7 = (1 + ¥v 2)3 & has 
six particular cognate forms all unequal. Should 7; = (2 + v 3)! 
+ (2 — ¥v 3) (2 + y 3)3, & has six particular cognate forms, but 
only three unequal, each of the unequal forms occurring twice. 
§10. Proposition I. An algebraical expression 7; can always be 
brought to a simple state. 
1 
For r; may be cleared of all surds such as 1 ™ having a rational 
1 
— 
value. Suppose that 7 then involves a surd Ae , not a root of unity, 
by means of which an equation such as (3) can be formed. Substitute 
1 
Mm. ° e ° = » 
for 4, in 7, its value e; as thus given. The result will be to elimi- 
1 
™ . . . . . 
nate 4, from 7, without introducing into the expression any new 
1 
. . ™m ° 
surd as high in rank as 4 , and at the same time not a root of 
unity. By continuing to make all the eliminations of this kind that 
are possible, we at last reach a point where no equation of the type 
(3) can any longer be formed. Then because, by the course that has 
1 
been pursued, no roots of the form 1” having a rational value have 
been left in 71, 71 is in a simple state. 
S11. It is known that, if W be any whole number, the equation 
whose roots are the primitive Vt roots of unity is rational and 
irreducible. 
§12. Let WV be the continued product of the distinct prime numbers 
n, a, 6, etc. Let w be a primitive nt root of unity, 6; a primitive 
a‘h yoot of unity, and so on. Let w represent any one iodifferently 
of the primitive nth roots of unity, 0 any one indifferently of the 
primitive at® roots of unity, and so on. Let f(w1, 0, etc.,) be a 
rational function of 1, 0), etc. Then a corollary from §11 is, that if 
J (#1, 91, ete.) = 0, f(w, 0, etc.) = 0. For t; being a primitive VW 
root of unity, and ¢ representing any one indifferently of the primitive 
Nth roots of unity, we may put 
7 N—1 N—2 
J (1, (1, etc. ) = ahi + daogt) + ete. = 0, 
= N—2 
and f(w, 0, etc,) = at ce waat + ete. ; 
where the coefficients 1, a, etc., are rational. Should these coefti- 
cients be all zero, f(w, 0, etc.) = 0. Should they not be all zero, let 
a, be the first that is not zero. Then we may put 
f(w1, 1, etc.) = ar { ¢ (4) } = att” + ete = 0. 
