Proving two field extensions aren't equal for different cube roots

Question

Let $ d_1, d_2$ be square-free integers $\neq 0, \pm 1$, with $ d_1 \neq \pm d_2 $. We can assume that $ x^3 - d_i $ and $ x^3 - d_i^2 $ are irreducible over $ \mathbb{Q}$. I've shown that $ \text{Tr}_{\mathbb{Q}(\sqrt[3]{d_i})/\mathbb{Q}}(\sqrt[3]{d_i}) = \text{Tr}_{\mathbb{Q}(\sqrt[3]{d_i})/\mathbb{Q}}(\sqrt[3]{d_i^2}) = 0 \text{ for } i = 1,2$. I now want to show $ \mathbb{Q}(\sqrt[3]{d_1}) \neq \mathbb{Q}(\sqrt[3]{d_2}).$ How do I do this?