If `alpha` Is the fifth root of unity, then :

Options

A. `|1+alpha+alpha^(2)+alpha^(3)+alpha^(4)|=0`
B. `|1+alpha+alpha^(2)+alpha^(3)|=1`
C. `|1+alpha+alpha^(2)|=2cos"(pi)/(5)`
D. `|1+alpha|=2cos"(pi)/(10)`

Explanation

To solve the problem, we need to analyze the properties of the fifth roots of unity. The fifth roots of unity are the solutions to the equation \( z^5 = 1 \). These roots can be expressed in the form:

\[ \alpha_k = e^{2\pi i k / 5} \quad \text{for } k = 0, 1, 2, 3, 4 \]

In this case, we can take \( \alpha = e^{2\pi i / 5} \).

### Step 1: Write the expression for the fifth root of unity
The fifth root of unity can be expressed as:
\[ \alpha = \cos\left(\frac{2\pi}{5}\right) + i \sin\left(\frac{2\pi}{5}\right) \]

### Step 2: Use the property of roots of unity
The sum of all the fifth roots of unity is given by:
\[ 1 + \alpha + \alpha^2 + \alpha^3 + \alpha^4 = 0 \]
This implies:
\[ \alpha + \alpha^2 + \alpha^3 + \alpha^4 = -1 \]

### Step 3: Find the modulus of the sum
We need to find the modulus of \( 1 + \alpha + \alpha^2 + \alpha^3 + \alpha^4 \):
\[ |1 + \alpha + \alpha^2 + \alpha^3 + \alpha^4| = |0| = 0 \]

### Step 4: Analyze the modulus of partial sums
Now, consider the modulus of \( 1 + \alpha + \alpha^2 + \alpha^3 \):
\[ |1 + \alpha + \alpha^2 + \alpha^3| = |-\alpha^4| \]
Since \( |\alpha^4| = 1 \) (as all roots of unity lie on the unit circle), we have:
\[ |1 + \alpha + \alpha^2 + \alpha^3| = 1 \]

### Step 5: Further analysis of the modulus
Next, we analyze \( |1 + \alpha + \alpha^2| \):
\[ |1 + \alpha + \alpha^2| = |-\alpha^3 - \alpha^4| \]
Factoring out \( -\alpha^3 \):
\[ |1 + \alpha + \alpha^2| = |\alpha^3| \cdot |1 + \alpha| = |1 + \alpha| \]
Since \( |\alpha^3| = 1 \), we have:
\[ |1 + \alpha + \alpha^2| = |1 + \alpha| \]

### Step 6: Calculate the modulus explicitly
Now we compute \( |1 + \alpha| \):
\[ |1 + \alpha| = |1 + \cos\left(\frac{2\pi}{5}\right) + i \sin\left(\frac{2\pi}{5}\right)| \]
This can be simplified using the formula for the modulus:
\[ |1 + \alpha| = \sqrt{(1 + \cos\left(\frac{2\pi}{5}\right))^2 + \sin^2\left(\frac{2\pi}{5}\right)} \]
Using the identity \( \sin^2\theta + \cos^2\theta = 1 \):
\[ = \sqrt{(1 + \cos\left(\frac{2\pi}{5}\right))^2 + (1 - \cos^2\left(\frac{2\pi}{5}\right))} \]
This simplifies to:
\[ = \sqrt{2 + 2\cos\left(\frac{2\pi}{5}\right)} = 2\cos\left(\frac{\pi}{5}\right) \]

### Conclusion
Thus, we conclude that:
1. \( 1 + \alpha + \alpha^2 + \alpha^3 + \alpha^4 = 0 \)
2. \( |1 + \alpha + \alpha^2 + \alpha^3| = 1 \)
3. \( |1 + \alpha + \alpha^2| = 2\cos\left(\frac{\pi}{5}\right) \)

### Final Answer
Options 1, 2, and 3 are correct.