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?

Start with $b \sqrt[3]{2} + c \sqrt[3]{4} = -a$. Cube this relation to find another
equation of the form $B \sqrt[3]{2} + C \sqrt[3]{4} = -A$ for rationals A, B, C. Eliminating
the cube root of 4 from these equations will tell you that the cube root of 2 is rational.
This contradiction shows that $a = b = c = 0$.
