617. The Cube Roots of Unity.
Suppose x = 3 1 ; then x3 = 1, or x3 — 1 = 0; that is (x - 1)(x2 + x + 1) = 0. either x — 1 = 0, or x2 + x + 1 = 0; whence x=1, or x =
It may be shown by actual involution that each of these values when cubed is equal to unity. Thus unity has three cube roots,
-14-V33 -1-V33 ' 2 ' 2 '
two of which are imaginary expressions.
Let us denote these by a and b ; then since they are the roots of the equation x^2 + x + 1 = 0, their product is equal to unity ;
that is, ab = 1 a3b = a2; that is, b = a^2, since a3 = 1.
Similarly we may show that a = b^2.
618. Since each of the imaginary roots is the square of the other, it is usual to denote the three cube roots of unity by 1, \omega, \omega^2 * Also (X) satisfies the equation x2 + x + 1 = 0;
\omega^2 + \omega + 1 = 0;
that is, the sum of the three cube roots of unity is zero. Again \omega \omega^2 = \omega^3 = 1 ;
therefore (1) the product of the two imaginary roots is unity;
(2) every integral power of \omega^3 is unity.
- The Greek letter Omega.
