If If ` 1, alpha_(1), alpha_(2), alpha_(3) , . . . alpha_(n - 1)` are the n nth roots of unity and n is odd or even natural numbers, then `(1 + alpha_(1)) (1 + alpha_(2)) . . . ( 1 = alpha _(n- 1)) ` equals

Options

A. 1
B. `-1`
C. 0
D. none of these

Answer

To solve the problem, we need to evaluate the product \( (1 + \alpha_1)(1 + \alpha_2) \cdots (1 + \alpha_{n-1}) \) where \( 1, \alpha_1, \alpha_2, \ldots, \alpha_{n-1} \) are the \( n \)-th roots of unity.

### Step-by-Step Solution:

1. **Understanding the Roots of Unity**:
The \( n \)-th roots of unity are given by the equation \( z^n = 1 \). The solutions to this equation are:
\[ z = 1, \alpha_1, \alpha_2, \ldots, \alpha_{n-1} \]
where \( \alpha_k = e^{2\pi i k/n} \) for \( k = 0, 1, 2, \ldots, n-1 \).

2. **Factoring the Polynomial**:
The polynomial whose roots are the \( n \)-th roots of unity can be expressed as:
\[ z^n - 1 = (z - 1)(z - \alpha_1)(z - \alpha_2) \cdots (z - \alpha_{n-1}) \]

3. **Finding the Product**:
We want to evaluate the product \( (1 + \alpha_1)(1 + \alpha_2) \cdots (1 + \alpha_{n-1}) \). This can be rewritten in terms of the polynomial:
\[ P(z) = z^n - 1 \]
We can find this product by substituting \( z = -1 \):
\[ P(-1) = (-1)^n - 1 \]

4. **Calculating \( P(-1) \)**:
- If \( n \) is even, \( (-1)^n = 1 \), so:
\[ P(-1) = 1 - 1 = 0 \]
- If \( n \) is odd, \( (-1)^n = -1 \), so:
\[ P(-1) = -1 - 1 = -2 \]

5. **Relating \( P(-1) \) to the Product**:
We can relate \( P(-1) \) to the product we want:
\[ P(-1) = (-1 - \alpha_1)(-1 - \alpha_2) \cdots (-1 - \alpha_{n-1}) = (-1)^{n-1} (1 + \alpha_1)(1 + \alpha_2) \cdots (1 + \alpha_{n-1}) \]
Therefore:
\[ (1 + \alpha_1)(1 + \alpha_2) \cdots (1 + \alpha_{n-1}) = \frac{P(-1)}{(-1)^{n-1}} \]

6. **Final Evaluation**:
- For even \( n \):
\[ (1 + \alpha_1)(1 + \alpha_2) \cdots (1 + \alpha_{n-1}) = \frac{0}{(-1)^{n-1}} = 0 \]
- For odd \( n \):
\[ (1 + \alpha_1)(1 + \alpha_2) \cdots (1 + \alpha_{n-1}) = \frac{-2}{(-1)^{n-1}} = 2 \]

### Conclusion:
Thus, the final result is:
- If \( n \) is even, the product equals \( 0 \).
- If \( n \) is odd, the product equals \( 2 \).