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 evaluate the expression \( \log_{2} |1 + \alpha + \alpha^{2} + \alpha^{3} - \frac{1}{\alpha}| \) where \( \alpha \) is an imaginary fifth root of unity.

### Step-by-Step Solution:

1. **Understanding 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 roots are \( \omega, \omega^2, \omega^3, \omega^4 \).

2. **Using the Property of Roots of Unity**:
Since \( \alpha \) is an imaginary fifth root of unity, we know:
\[ 1 + \alpha + \alpha^2 + \alpha^3 + \alpha^4 = 0 \]
This implies:
\[ 1 + \alpha + \alpha^2 + \alpha^3 = -\alpha^4 \]

3. **Substituting into the Expression**:
We substitute \( 1 + \alpha + \alpha^2 + \alpha^3 \) into our original 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 rewrite \( -\frac{1}{\alpha} \) with 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:
\[ |-\frac{2}{\alpha}| = \frac{2}{|\alpha|} \]
Since \( |\alpha| = 1 \) (as it is a root of unity), we have:
\[ |-\frac{2}{\alpha}| = 2 \]

6. **Calculating the Logarithm**:
Now we can compute the logarithm:
\[ \log_{2} |1 + \alpha + \alpha^{2} + \alpha^{3} - \frac{1}{\alpha}| = \log_{2}(2) \]
Since \( \log_{2}(2) = 1 \), we conclude that:
\[ \log_{2} |1 + \alpha + \alpha^{2} + \alpha^{3} - \frac{1}{\alpha}| = 1 \]

### Final Answer:
\[ \log_{2} |1 + \alpha + \alpha^{2} + \alpha^{3} - \frac{1}{\alpha}| = 1 \]