Prove $\mathbb{Q}(2^{1/4})/ \mathbb{Q}$ is NOT a Galois extension.

Question

I'm trying to show that, if $K/L$ and $L/F$ are Galois extensions, this doesn't necessarily imply that $K/F$ is also a Galois extension.

I have already shown that $\mathbb{Q}(2^{1/4})/ \mathbb{Q}(\sqrt 2)$ and $\mathbb{Q}(\sqrt 2) / \mathbb Q$ are Galois extensions. To finish the proof, I just need to show that $\mathbb{Q}(2^{1/4})/ \mathbb{Q}$ is NOT a Galois extension. But how do I show it?

Answer 1

Galois extensions are normal. This means that every polynomial either splits completely in the extension or has no roots in the extension. For your case, take the polynomial $x^4-2$. It can't split completely, because it has imaginary roots and your field is contained in $\mathbb{R}$. But it also has one root.

Answer 2

The polynomial $x^4-2$ has a root in $\mathbb{Q}\left(\sqrt[4]2\right)$, but not all of its roots belong to it.