Trying this for a while and, I found out that if there is another solution $(a_0,b_0,c_0)$, then there is an infinite amount of solutions $(a_0/2,b_0/2,c_0/2), (a_0/4,b_0/4,c_0/4) \ldots$, but I am not sure that this is a contradiction.
Can someone suggest a new insight or help to prove this statement?
2026-04-02 10:51:22.1775127082
Prove that if $a+b\sqrt[3]{2} + c\sqrt[3]{4} = 0$ then $a=b=c=0$, where $a,b,c$ are rational numbers
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Hint: If $a + b\sqrt[3]{2} + c\sqrt[3]{4} = 0$, then $\sqrt[3]{2}$ is a solution to $a + bx + cx^2 = 0$. What do we know about quadratic equations with rational coefficients?