Let $\alpha=\sqrt[3] 2$, $\beta=\sqrt[4] 5$.
I would hope to show that $c_0+c_1\alpha+c_2\alpha^2=0$ is impossible, where $c_i$ are elements in the field $\mathbb{Q}(\beta)$, and $c_2$ is nonzero. How do I go about proving it? Thanks!
What I tried is the quadratic formula, but got stuck since the square root operation does not "remain in the field".