Suppose $m$ is the minimal polynomial of a set $S$ of complex numbers such that for a complex number $\alpha\not\in S$, $\alpha$ is a root of the derivative of $m$. I want to prove that $[\mathbb{Q}(m(\alpha)):\mathbb{Q}]\leq[\mathbb{Q}(\alpha):\mathbb{Q}]$. I have tried some examples and my intuition tells me it has something to do with linear combinations of $\alpha$ being already in the extension, but I don't really know how to prove it. Any help or hint will be welcome. Thank you very much!
PS: This is in the context of Riemann surfaces, but I'm not sure if that is relevant.