To solve the problem of finding the number of roots of unity that are also roots of the equation \( z^2 + az + b = 0 \) for some integers \( a \) and \( b \), we can follow these steps:
### Step 1: Understanding Roots of Unity
A root of unity is a complex number \( \omega \) that satisfies the equation \( z^n = 1 \) for some positive integer \( n \). The roots of unity can be expressed in the form:
\[
\omega = e^{2\pi i k/n}
\]
for \( k = 0, 1, 2, \ldots, n-1 \).
### Step 2: Setting Up the Quadratic Equation
Given that \( \omega \) is a root of the quadratic equation \( z^2 + az + b = 0 \), we can substitute \( \omega \) into the equation:
\[
\omega^2 + a\omega + b = 0
\]
### Step 3: Properties of Roots of Quadratic Equations
From the properties of quadratic equations, if \( \omega \) is a root, then its conjugate \( \overline{\omega} \) is also a root. Therefore, we can use the relationships:
- The sum of the roots \( \omega + \overline{\omega} = -a \)
- The product of the roots \( \omega \cdot \overline{\omega} = b \)
### Step 4: Modulus of Roots
Since \( \omega \) is a root of unity, we know that \( |\omega| = 1 \). Thus, we can express:
\[
\omega \cdot \overline{\omega} = |\omega|^2 = 1 \implies b = 1
\]
### Step 5: Finding \( a \)
Next, we find \( a \) using the sum of the roots:
\[
\omega + \overline{\omega} = -a
\]
The sum \( \omega + \overline{\omega} = 2 \cos(\theta) \) where \( \theta \) is the argument of \( \omega \). Thus:
\[
-a = 2 \cos(\theta) \implies a = -2 \cos(\theta)
\]
This means \( a \) can take values in the range \( [-2, 2] \).
### Step 6: Possible Values of \( a \)
The integer values of \( a \) that satisfy \( -2 \leq a \leq 2 \) are:
\[
a = -2, -1, 0, 1, 2
\]
### Step 7: Finding the Corresponding Quadratic Equations
For each integer value of \( a \), we can form the corresponding quadratic equations:
1. \( z^2 - 2z + 1 = 0 \) (roots: \( 1 \))
2. \( z^2 - z + 1 = 0 \) (roots: \( \frac{1}{2} \pm i \frac{\sqrt{3}}{2} \))
3. \( z^2 + 1 = 0 \) (roots: \( i, -i \))
4. \( z^2 + z + 1 = 0 \) (roots: \( -\frac{1}{2} \pm i \frac{\sqrt{3}}{2} \))
5. \( z^2 + 2z + 1 = 0 \) (roots: \( -1 \))
### Step 8: Counting the Roots of Unity
Now, we count the roots of unity among these equations:
1. From \( z^2 - 2z + 1 = 0 \): \( 1 \) (1 root)
2. From \( z^2 - z + 1 = 0 \): \( \frac{1}{2} \pm i \frac{\sqrt{3}}{2} \) (2 roots)
3. From \( z^2 + 1 = 0 \): \( i, -i \) (2 roots)
4. From \( z^2 + z + 1 = 0 \): \( -\frac{1}{2} \pm i \frac{\sqrt{3}}{2} \) (2 roots)
5. From \( z^2 + 2z + 1 = 0 \): \( -1 \) (1 root)
### Total Count
Adding these up gives:
\[
1 + 2 + 2 + 2 + 1 = 8
\]
Thus, the total number of roots of unity that are also roots of the quadratic equation is **8**.
### Final Answer
The number of roots of unity that are also the roots of the equation \( z^2 + az + b = 0 \) for some integers \( a \) and \( b \) is **8**.
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