If a, b, c are integers not all simultaneously equal then the minimum value of $|a + bω + {cω}^{2}|$ is (where $ω$ is the cube root of unity $\neq 1$ )

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$\frac{1}{2}$

$\frac{\sqrt{3}}{2}$

answer is D.

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Detailed Solution

$\begin{array}{rcl} {|a + bw + {cw}^{2}|}^{2} & = & (a + bw + w^{2}) (a + b w + c w^{2}) \\ = & a^{2} + b^{2} + c^{2} - ab - bc - ca \\ = & \frac{1}{2} ({(a - b)}^{2} + {(b - c)}^{2} + {(c - a)}^{2}) > 0 \end{array}$

a, b, c are integers not all equal min value =1

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