Understanding the Properties of Imaginary Cube Roots of Unity

[th4450]

Homework Statement
Let the roots of x² + x + 1 = 0 be $\alpha$ and $\beta$, form a quadratic equation with roots:
(i) $\alpha$², $\beta$²; and
(ii) $\frac{1}{\alpha}$, $\frac{1}{\beta}$.The attempt at a solution
sum of roots = $\alpha$ + $\beta$ = -1
product of roots = $\alpha\beta$ = 1

(i)
sum of roots = $\alpha$² + $\beta$² = (-1)² - 2(1) = -1
product of roots = $\alpha$²$\beta$² = 1² = 1
Quadratic equation: x² + x + 1 = 0

(ii)
sum of roots = $\frac{1}{\alpha}$ + $\frac{1}{\beta}$ = $\frac{-1}{1}$ = -1
product of roots = ($\frac{1}{\alpha}$)($\frac{1}{\beta}$) = 1
Quadratic equation: x² + x + 1 = 0

Is this weird? Did I do something wrong?
Thanks.

 

[pcm]

correct.
roots of original equations are imaginary cube roots of unity.
which has the property that the square of first root is equal to second root,and square of second root is equal to first root.
the same property for taking reciprocals.