Finding all intermediate fields STATEMENT: Let $\alpha$ be the real positive fourth root of 2. Find all intermediate fields in the extension $\mathbb{Q}(\alpha)$ of $\mathbb{Q}$.
QUESTION: I basically used the tower law to show that $\mathbb{Q}(\alpha),\mathbb{Q}(\alpha^2),\mathbb{Q}$ are all intermediate fields by tower law, and $\mathbb{Q}(\alpha^3)=\mathbb{Q}(\alpha)$.  I am just not sure how to show that these are all of the intermediate fields. I would really appreciate a hint or suggestion.

Are you familiar with the Galois Correspondence? Because you could compute the Galois group of the polynomial $x^4-2$ and write all its subgroups, and then by the correspondence you would obtain all the subfields between $\mathbb{Q}(\sqrt[4]{2},i)$ (the splitting field of the previous polynomial) and $\mathbb{Q}$: in particular one of the ramifications would be the subfields of $\mathbb{Q}\sqrt[4]{2}) \subset \mathbb{Q}(\sqrt[4]{2},i)$,
