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`

Explanation

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

### Step 1: Identify the cube roots of unity
The cube roots of unity are the solutions to the equation \(x^3 = 1\). They are given by:
\[ 1, \omega, \omega^2 \]
where \(\omega = e^{2\pi i / 3} = -\frac{1}{2} + i\frac{\sqrt{3}}{2}\) and \(\omega^2 = e^{4\pi i / 3} = -\frac{1}{2} - i\frac{\sqrt{3}}{2}\).

### Step 2: Assign values to \(\omega_1\) and \(\omega_2\)
Let:
\[ \omega_1 = \omega = -\frac{1}{2} + i\frac{\sqrt{3}}{2} \]
\[ \omega_2 = \omega^2 = -\frac{1}{2} - i\frac{\sqrt{3}}{2} \]

### Step 3: Calculate \(\omega_1 - \omega_2\)
Now, we compute:
\[ \omega_1 - \omega_2 = \left(-\frac{1}{2} + i\frac{\sqrt{3}}{2}\right) - \left(-\frac{1}{2} - i\frac{\sqrt{3}}{2}\right) \]
This simplifies to:
\[ \omega_1 - \omega_2 = i\frac{\sqrt{3}}{2} + i\frac{\sqrt{3}}{2} = i\sqrt{3} \]

### Step 4: Calculate \((\omega_1 - \omega_2)^2\)
Next, we find:
\[ (\omega_1 - \omega_2)^2 = (i\sqrt{3})^2 = i^2 \cdot 3 = -1 \cdot 3 = -3 \]

### Final Answer
Thus, we have:
\[ (\omega_1 - \omega_2)^2 = -3 \]