$F=\mathbb{Q}(\alpha_1,\alpha_2, ... , \alpha_n)$, where $\alpha_i^2 \in \mathbb{Q}$ implies $\sqrt[3]{2} \notin F$
Showing that $\sqrt[3]{2}\notin\Bbb Q(\alpha_1,...,\alpha_k)$ where $\alpha_i^2\in\Bbb Q\ \forall i$