If `omega` is a cube root of unity but not equal to 1, then minimum value of `abs(a+bomega+comega^(2))`, (where a,b and c are integers but not all equal ), is

Options

A. 0
B. `sqrt(3)/2`
C. 1
D. 2

Answer

To find the minimum value of \( |a + b\omega + c\omega^2| \), where \( \omega \) is a cube root of unity (not equal to 1) and \( a, b, c \) are integers but not all equal, we can follow these steps:

### Step 1: Understand the properties of cube roots of unity
The cube roots of unity are given by:
\[ \omega = e^{2\pi i / 3} = -\frac{1}{2} + i\frac{\sqrt{3}}{2} \]
and
\[ \omega^2 = e^{-2\pi i / 3} = -\frac{1}{2} - i\frac{\sqrt{3}}{2} \]
We also know that:
\[ 1 + \omega + \omega^2 = 0 \quad \text{or} \quad \omega^2 = -1 - \omega \]

### Step 2: Rewrite the expression
We need to find the minimum value of:
\[ |a + b\omega + c\omega^2| \]
Using the property \( \omega^2 = -1 - \omega \), we can rewrite the expression:
\[ |a + b\omega + c(-1 - \omega)| = |(a - c) + (b - c)\omega| \]

### Step 3: Set new variables
Let:
\[ x = a - c \quad \text{and} \quad y = b - c \]
Thus, we need to minimize:
\[ |x + y\omega| \]

### Step 4: Calculate the modulus
The modulus can be expressed as:
\[ |x + y\omega| = |x + y\left(-\frac{1}{2} + i\frac{\sqrt{3}}{2}\right)| \]
This can be simplified to:
\[ |x - \frac{y}{2} + i\frac{y\sqrt{3}}{2}| \]
Using the formula for the modulus of a complex number:
\[ |z| = \sqrt{(\text{Re}(z))^2 + (\text{Im}(z))^2} \]
we have:
\[ |x - \frac{y}{2} + i\frac{y\sqrt{3}}{2}| = \sqrt{\left(x - \frac{y}{2}\right)^2 + \left(\frac{y\sqrt{3}}{2}\right)^2} \]

### Step 5: Simplify the expression
This simplifies to:
\[ \sqrt{\left(x - \frac{y}{2}\right)^2 + \frac{3y^2}{4}} \]
To minimize this expression, we can analyze the values of \( x \) and \( y \).

### Step 6: Consider integer values
Since \( a, b, c \) are integers and not all equal, we can consider simple cases. For instance, let’s take:
- \( a = 0, b = 1, c = 0 \) gives \( |0 + 1\omega + 0| = |\omega| = 1 \)
- \( a = 0, b = -1, c = 0 \) gives \( |0 - 1\omega + 0| = |-\omega| = 1 \)

### Step 7: Conclusion
The minimum value of \( |a + b\omega + c\omega^2| \) occurs when \( |b| = 1 \) and \( c = 0 \) or similar configurations, leading to:
\[ \text{Minimum value} = 1 \]

### Final Answer
The minimum value of \( |a + b\omega + c\omega^2| \) is \( \boxed{1} \).