If `alpha` is an imaginary fifth root of unity, then `log_(2)|1+alpha+alpha^(2)+alpha^(3)-1/alpha|=`

Options

A. 1
B. 0
C. 2
D. `-1`

Explanation

To solve the problem, we need to find the value of \( \log_2 \left| 1 + \alpha + \alpha^2 + \alpha^3 - \frac{1}{\alpha} \right| \) where \( \alpha \) is an imaginary fifth root of unity.

### Step-by-Step Solution:

1. **Understand the Fifth Roots of Unity**:
The fifth roots of unity are the solutions to the equation \( x^5 = 1 \). They can be expressed as:
\[ 1, \omega, \omega^2, \omega^3, \omega^4 \]
where \( \omega = e^{2\pi i / 5} \). The imaginary fifth roots of unity are \( \omega, \omega^2, \omega^3, \omega^4 \).

2. **Sum of Roots**:
The sum of all fifth roots of unity is zero:
\[ 1 + \omega + \omega^2 + \omega^3 + \omega^4 = 0 \]
Therefore, we can express the sum \( 1 + \alpha + \alpha^2 + \alpha^3 \) as:
\[ 1 + \alpha + \alpha^2 + \alpha^3 = -\alpha^4 \]

3. **Substituting into the Expression**:
Now substitute this into our expression:
\[ 1 + \alpha + \alpha^2 + \alpha^3 - \frac{1}{\alpha} = -\alpha^4 - \frac{1}{\alpha} \]

4. **Finding a Common Denominator**:
To combine the terms, we need a common denominator:
\[ -\alpha^4 - \frac{1}{\alpha} = -\alpha^4 - \frac{1}{\alpha} = -\frac{\alpha^5 + 1}{\alpha} \]
Since \( \alpha^5 = 1 \), we have:
\[ -\frac{1 + 1}{\alpha} = -\frac{2}{\alpha} \]

5. **Taking the Modulus**:
Now, we need to find the modulus:
\[ \left| -\frac{2}{\alpha} \right| = 2 \left| \frac{1}{\alpha} \right| = 2 \cdot 1 = 2 \]
(since \( |\alpha| = 1 \) for any root of unity).

6. **Calculating the Logarithm**:
Now we can substitute this modulus into the logarithm:
\[ \log_2 \left| 1 + \alpha + \alpha^2 + \alpha^3 - \frac{1}{\alpha} \right| = \log_2(2) \]

7. **Final Result**:
Since \( \log_2(2) = 1 \), we conclude that:
\[ \log_2 \left| 1 + \alpha + \alpha^2 + \alpha^3 - \frac{1}{\alpha} \right| = 1 \]

### Final Answer:
\[ \boxed{1} \]