Suppose `omega_(1)` and `omega_(2)` are two distinct cube roots of unity different from 1. Then, what is `( omega_(1) - omega_(2))^(2)` equal to ?

Options

A. 3
B. 1
C. `-1`
D. `-3`

Answer

To solve the problem, we need to find the value of \((\omega_1 - \omega_2)^2\) where \(\omega_1\) and \(\omega_2\) are two distinct cube roots of unity different from 1.

### Step-by-Step Solution:

1. **Identify the cube roots of unity**:
The cube roots of unity are given by:
\[ 1, \omega, \omega^2 \]
where:
\[ \omega = e^{2\pi i / 3} = -\frac{1}{2} + i\frac{\sqrt{3}}{2} \]
\[ \omega^2 = e^{4\pi i / 3} = -\frac{1}{2} - i\frac{\sqrt{3}}{2} \]
Here, \(\omega_1 = \omega\) and \(\omega_2 = \omega^2\).

2. **Calculate \(\omega_1 - \omega_2\)**:
Substitute the values of \(\omega_1\) and \(\omega_2\):
\[ \omega_1 - \omega_2 = \left(-\frac{1}{2} + i\frac{\sqrt{3}}{2}\right) - \left(-\frac{1}{2} - i\frac{\sqrt{3}}{2}\right) \]
Simplifying this gives:
\[ \omega_1 - \omega_2 = -\frac{1}{2} + i\frac{\sqrt{3}}{2} + \frac{1}{2} + i\frac{\sqrt{3}}{2} = i\sqrt{3} \]

3. **Square the result**:
Now we need to find \((\omega_1 - \omega_2)^2\):
\[ (\omega_1 - \omega_2)^2 = (i\sqrt{3})^2 \]
This simplifies to:
\[ (i\sqrt{3})^2 = i^2 \cdot 3 = -1 \cdot 3 = -3 \]

4. **Final answer**:
Therefore, \((\omega_1 - \omega_2)^2 = -3\).

### Conclusion:
The value of \((\omega_1 - \omega_2)^2\) is \(-3\).