An introduction to the modern theory of equations, by Florian Cajori.

CHAPTER VI

SOLUTION OF BINOMIAL EQUATIONS AND RECIPROCAL
EQUATIONS
63. The Binomial Equation.
X11 - a = 0O
where a is either real or complex, may be solved trigonometric
cally as follows. Let
t == a = r cos (2 kr + 0) + i sin (2 k7r + 0) f,
where k may assume any integral value. Then, by De Moivre's
Theorem,
r 22 k7~r+. 2kr+0 ]
x- r cos — + i sin- ~
L it n J
By assigning to k any n consecutive integral values we obtain
n values for x and no more than n, since the n values recur in
periods.
It is readily seen that the roots are all complex when a is a
2 1krr + 0
complex number. For, to obtain a real root, + must be
zero or a multiple of 7r; that is, 2 ckr+ 0 will be zero or a
multiple of 7r; hence a itself must be real, which is contrary
to supposition.
When a = +1,
then x" = 1 = cos 2 kIr + i sin 2 k7r,
2 kr. 2 k7
and x = cos — + - sin --, I
n
74