I would like to find a simple Galois extension $\mathbb{Q}(\alpha )/\mathbb{Q}$ whose order is odd and has at laest two different prime divisors (i.e. $|Gal(\mathbb{Q}(\alpha)/\mathbb{Q})|=[\mathbb{Q}(\alpha ):\mathbb{Q}]$ is odd and it is not of the form $p^k$ for some $p\in \mathbb{P},k \in \mathbb{N} $) such that:
$Gal(\mathbb{Q}(\alpha)/\mathbb{Q}) \neq \bigcup_{\mathbb{Q}\subsetneq M\subseteq\mathbb{Q}(\alpha)} Gal(\mathbb{Q}(\alpha)/M)$
That is, there is an automorphism $\sigma \in Gal(\mathbb{Q}(\alpha)/\mathbb{Q})$ and a sub field $\mathbb{Q}\subsetneq M\subseteq\mathbb{Q}(\alpha)$ such that $\sigma|_M\neq id$.
Somehow it seems to me like the answer should be obvious but I cant see it,
Thanks in advance.