I'm looking at a proof of the following:
Let $K = \mathbb Q(\alpha)$ and $\beta \in K$ with minimal polynomial $g \in \mathbb Q[X]$, where the roots of $g$ are $\beta_1, \ldots , \beta_m$. Then $d_i = |\{\sigma : K \hookrightarrow \mathbb C \ | \ \sigma(\beta) = \beta_i \}|$ is independent of $i$.
The proof starts by considering $h$, the minimal polynomial of $\alpha$ over $\mathbb Q(\beta)$, and says that $d_i \leq \mathrm{deg}(h) = [\mathbb Q(\alpha) : \mathbb Q (\beta)]$.
Why is this true? I did understand this at one point...
Thanks