How to make a field extension chain

Question

I'm looking for $\alpha$ such that $\mathbb{Q}\subset \mathbb{Q}(\sqrt{17})\subset \mathbb{Q}(\alpha)\subset \mathbb{Q}(\zeta_{17}+\zeta_{17}^{-1})\subset \mathbb{Q}(\zeta_{17})$ will be a chain of extensions with extension degree of 2 for every extension, when $\zeta_{17}=e^{2\pi i/17}$. How do I approach that? (need a direction, not solution. thanks)