If omega is a cube root of unity but not equa | vedclass

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(C) Let $y = |a + b\omega + c\omega^2|$.For $y$ to be minimum,$y^2$ must be minimum.$y^2 = |a + b\omega + c\omega^2|^2 = (a + b\omega + c\omega^2)(a +

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(C) Let $y = |a + b\omega + c\omega^2|$.For $y$ to be minimum,$y^2$ must be minimum.$y^2 = |a + b\omega + c\omega^2|^2 = (a + b\omega + c\omega^2)(a + b\bar{\omega} + c\bar{\omega}^2)$.Using $\omega^2 = \bar{\omega}$ and $\omega = \bar{\omega}^2$,we get $y^2 = a^2 + b^2 + c^2 - ab - bc - ca$.This can be written as $y^2 = \frac{1}{2}[(a - b)^2 + (b - c)^2 + (c - a)^2]$.Since $a, b, c$ are integers and not all equal,the minimum value occurs when two variables are equal and the third differs by $1$ (e.g.,$a=0, b=0, c=1$).Substituting these values,$y^2 = \frac{1}{2}[(0-0)^2 + (0-1)^2 + (1-0)^2] = \frac{1}{2}[0 + 1 + 1] = 1$.Thus,the minimum value of $y$ is $\sqrt{1} = 1$.

If $\omega$ is a cube root of unity but not equal to $1$,then the minimum value of $|a + b\omega + c\omega^2|$ (where $a, b, c$ are integers but not all equal) is

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