Let $L/\mathbb Q$ be a (finite) Galois extension of degree $n$ with Galois group $\Gamma$. We know that there is a primitive element or generator $\alpha$ of this extension.
My question:
Is the number of Galois conjugates of $\alpha$ (i.e. the length of the orbit of $\alpha$ under $\Gamma$) smaller than or equal to the order of the Galois group?
And under which conditions are they the same?
What does the Orbit-Stabilizer theorem tell us? If $\Gamma = \textrm{Gal}(L/\mathbb{Q})$, then because the extension is finite, $$ |\Gamma | = |\Gamma \cdot \alpha| \cdot | \textrm{Stab}_{\Gamma}(\alpha)|. $$ In particular, the number $|\Gamma \cdot \alpha|$ of Galois conjugates of $\alpha$ is less than or equal to the order $|\Gamma|$ of the Galois group. (It will, in fact, be an equality: the stabilizer of $\alpha$ must be trivial, else $\alpha$ would not be a generator.)