<s>[INST] If $\omega$ is a cube root of unity $\not = 1$ then find the minimum value of $|a+b\omega +c\omega^2|$, where $a,b,c$ are integers but not all equal. Let $z=a+b\omega + c\omega^2$
$$z=a+b\omega -c (1+\omega)$$
$$z=a-c+\omega (b-c)$$
Therefore $$|a-c+\omega (b-c)| \ge ||a-c|-|b-c||$$
How should I proceed?
 [/INST] We want to minimize $|x+y\omega|^2=\bigg(x-\frac{y}{2}\bigg)^2+\frac{3}{4}y^2$ where $x$ and $y$ are nonzero integers.
If $y$ is even, we can take $x=\frac{y}{2}$ and hence $|x+y\omega|^2 \geq \frac{3}{4}y^2 \geq 3$.
If $y$ is odd, we can take $x=\frac{y+1}{2}$ and hence $|x+y\omega|^2 \geq \frac{1}{4}+\frac{3}{4}y^2 \geq 1$.
Conclusion : the minimum is $1$. It is reached when $y=\pm 1,x=y$.
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