The normal closure of $\mathbb{Q}(\alpha)$ with degree a power of 2 is in $\mathbb{R}$

Question

In Galois theory I encountered the following question.

Let $\alpha \in \mathbb{R}$ and $L$ be the normal closure of $\mathbb{Q}(\alpha)$. Is $L \subset \mathbb{R}$ if $[L:\mathbb{Q}]$ is a power of 2?

I don't know how to prove this. Can I have a proof?

Answer 1

Hint: What happens if $\alpha=\sqrt[4]{2}$?