Problem 5
Clearly $ \mathbb{Q}(\alpha)\subset \mathbb{Q}(2^{\frac{1}{3}}) $. But also,
$ 2\alpha - \alpha = 16-17 \cdot 2^{\frac{1}{3}} $
So $ \mathbb{Q}(2^{\frac{1}{3}}) \subset \mathbb{Q}(\alpha) $ and $ \mathbb{Q}(2^{\frac{1}{3}}) = \mathbb{Q}(\alpha) $.
Thus $ 3=|\mathbb{Q}(2^{\frac{1}{3}}):\mathbb{Q}| = |\mathbb{Q}(\alpha):\mathbb{Q}| $.
