From the extension of my previous   For $a,b,c\in \mathbb{Q}$  \begin{align}  a + b \sqrt[3]{2}  + c\sqrt[3]{4} =0 \quad \Leftrightarrow \quad a=b=c=0  \end{align} again also one direction is easy, but how about other direction?
Given that $\displaystyle{a+b\sqrt[3]{2} +c\sqrt[3]{4} =0}$, where $a,b,c$ are integers. Show $a=b=c=0$  Do I use modular arithmetic?
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