Proving $M: \mathbb Q$ may not be a Galois Extension

Question

Let $M$ be a field and $ \mathbb Q \subset M \subset \mathbb Q(\zeta_n)$. I am trying to prove that the extension $M: \mathbb Q$ is not necessarily Galois, could anyone provide a counterexample? I know that the extension $\mathbb Q \subset \mathbb Q(\zeta_n)$ is galois since $\mathbb Q(\zeta_n)$ splits completely over $\mathbb Q$, but I'm not sure how to prove that $M: \mathbb Q$ is not guaranteed to be Galois...

Answer 1

$\Bbb Q(\zeta_n)$ has an Abelian Galois group, thus every subgroup is normal, thus every subextension is Galois.

Answer 2

Since $\mathbb{Q}(\zeta_n)$ is an abelian extension of $\mathbb{Q}$, Galois theory tells us that any intermediate extension $\mathbb{Q}\subset K\subset \mathbb{Q}(\zeta_n)$ must also be Galois over $\mathbb{Q}$