Given that $a+b\sqrt[3]{2} +c\sqrt[3]{4} =0$, where $a,b,c$ are integers. Show $a=b=c=0$ 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?

Let $\alpha=\sqrt[3]{2}$
Then $m_{\alpha}({x})=x^3-2$ is the minimal polynomial of $\alpha$ over $\Bbb{Q}$.
$f(x)=a+bx+cx^2$ is an anhilating polynomial of $\sqrt[3]{2}$ as $f(\alpha) =0$
Hence $m_{\alpha}(x) \mid f(x) $
$x^3-2\mid a+bx+cx^2$ implies $a=b=c=0$
