Counting the number of Galois conjugates While reading this solution set, I came across the following statement.  

In a Galois extension $K / F$, the number of Galois conjugates of any $\alpha \in K$ is equal to:
\begin{equation*}
\left| \text{Gal}(K/F) : \text{Gal}(K/F(\alpha)) \right| 
\end{equation*}
  and I am a bit perplexed where this number comes from.  

After thinking about this question for a bit, I realized that there is a "backdoor" approach to figuring this out.  We know that the minimal polynomial for $\alpha$, i.e. $m_{\alpha}(x)$, is equal to the squarefree part of the polynomial
\begin{equation*}
\prod_{\sigma \in \text{Gal}(K/F)} (x - \sigma(\alpha))
\end{equation*}
which has degree equal to the number of distinct Galois conjugates $\sigma(\alpha)$.  We also know that $\text{deg}(m_{\alpha}(x))$ is equal to $[F(\alpha): F]$.  But this is exactly what the index of $\text{Gal}(K/F)$ in $\text{Gal}(K/F(\alpha))$ is equal to, since:
\begin{align*}
\left| \text{Gal}(K/F) : \text{Gal}\big(K/F(\alpha) \big) \right|  & = \frac{ [K:F]}{[K:F(\alpha)]} \\[0.65em]
& = [F(\alpha):F]
\end{align*}
But I'm wondering if it is possible to see more "directly" that the number of (distinct) Galois conjugates of $\alpha$ is the number
\begin{equation*}
\left| \text{Gal}(K/F) : \text{Gal}\big(K/F(\alpha) \big) \right| 
\end{equation*}
 A: $\newcommand{\Gal}{\text{Gal}}$
Sometimes when $|A| = |B|$, it is because there is really an underlying bijection between $A$ and $B$.
Since $|\Gal(K/F) : \Gal(K/F(\alpha))|$ is the number of cosets of $\Gal(K/F(\alpha))$ in $\Gal(K/F)$, and this is the same as the number of Galois conjugates of $\alpha$, we might ask if we can establish a bijection between cosets of $\Gal(K/F(\alpha))$ in $\Gal(K/F)$ and conjugates of $\alpha$.
This is true, and we can see it as follows: Let $S$ be the set of conjugates of $\alpha$. $G$ acts on $S$ transitively, and the stabilizer of $\alpha \in S$ is the set of Galois elements fixing $\alpha$, equivalently, fixing $F(\alpha)$, i.e. $\Gal(K/F(\alpha))$, which establishes that $S$ and $\Gal(K/F)/\Gal(K/F(\alpha))$ are isomorphic as $\Gal(K/F)$-sets.
If you only care about the numbers, then you can use the Orbit-Stabilizer theorem to conclude that $|\Gal(K/F)| = |S||\Gal(K/F(\alpha))|$.
Edit: If you'd rather have something more concrete, the bijection between cosets and conjugates is given by $\sigma\Gal(K/F(\alpha)) \mapsto \sigma\alpha$. You can check this is a bijection in the same way you would check the stuff about group actions above.
