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Does there exist $v=(a,b,c)\in\mathbb{Q^3}$ with none of $v$'s terms being zero s.t. $ a+b\sqrt[3]2+c\sqrt[3]4=0$ ?

And I was doing undergraduate algebra 2 homework when I encountered it in my head. At first It seemed like it can be proved there can be no such $v$ like how $\sqrt{2}$, or $\sqrt{2}+\sqrt{3}$ are proved to be irrational, but this case wasn't easy like those. Or maybe I was too hasty.

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    Do you know field theory ?2012-09-19
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    Very little. I studied group theory in algebra 1 and now I'm in algebra 2. I have currently learned Euler,Fermat's theorems and what the field of fractions is.2012-09-19

3 Answers 3