Let $\alpha \in \overline{\mathbb{Q}}$ be an algebraic number so that $[\mathbb{Q}(\alpha) : \mathbb{Q} ] = N$, with $N$ a strictly positive integer. Then $A := \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ acts on the set of $N$ conjugates $\Omega(\alpha)$ of $\alpha$.
(I am sorry in advance for asking stupid questions.)
My first question is certainly well known: when does $A$ act transitively on $\Omega(\alpha)$ ?
Second question: let $\mathbb{Q}(\beta)$ be a not necessarily proper subfield of $\mathbb{Q}(\alpha)$.
What can we say about the size of $\beta^A$ in relation to $N$ ? Does it always divide $\vert \alpha^A \vert$ (or only with certainty if $A$ acts transitively on $\Omega(\alpha)$) ?
First question: giving a $\gamma\in\Omega(\alpha)$ is equivalent to giving an embedding $\mathbb Q(\alpha)\hookrightarrow\overline{\mathbb Q}$, by sending $\alpha$ to $\gamma$. Now standard arguments say such an embedding can be extended to an isomorphism $\overline{\mathbb Q}\to\overline{\mathbb Q}$, which we can call $\sigma$. Then $\sigma(\alpha)=\gamma$, so the action is transitive.
Second question: By the above question, the size of $\beta^A$ is simply the degree of the extension $[\mathbb Q(\beta):\mathbb Q]$. Thus in particular it divides the size of $\alpha^A$, which is the degree of the extension $[\mathbb Q(\alpha):\mathbb Q]$.