If the fourth roots of unity are `z_1, z_2, z_3, z_4 and z_1^2+z_2^2+z_3^2+z_4^2` is equal to :

Options

A. 1
B. 0
C. `i`
D. None of these

Answer

To solve the problem of finding \( z_1^2 + z_2^2 + z_3^2 + z_4^2 \) where \( z_1, z_2, z_3, z_4 \) are the fourth roots of unity, we can follow these steps:

### Step 1: Identify the fourth roots of unity
The fourth roots of unity are the solutions to the equation \( z^4 = 1 \). This can be rewritten as:
\[ z^4 - 1 = 0 \]
Factoring gives:
\[ (z^2 - 1)(z^2 + 1) = 0 \]
From \( z^2 - 1 = 0 \), we get:
\[ z^2 = 1 \implies z = 1, -1 \]
From \( z^2 + 1 = 0 \), we get:
\[ z^2 = -1 \implies z = i, -i \]
Thus, the fourth roots of unity are:
\[ z_1 = 1, \quad z_2 = -1, \quad z_3 = i, \quad z_4 = -i \]

### Step 2: Calculate the squares of the roots
Now we calculate the squares of each of these roots:
\[ z_1^2 = 1^2 = 1 \]
\[ z_2^2 = (-1)^2 = 1 \]
\[ z_3^2 = i^2 = -1 \]
\[ z_4^2 = (-i)^2 = (-1)^2 = -1 \]

### Step 3: Sum the squares
Now we sum these squares:
\[ z_1^2 + z_2^2 + z_3^2 + z_4^2 = 1 + 1 + (-1) + (-1) \]
Calculating this gives:
\[ 1 + 1 - 1 - 1 = 0 \]

### Conclusion
Thus, the value of \( z_1^2 + z_2^2 + z_3^2 + z_4^2 \) is:
\[ \boxed{0} \]