When branch points lie in $\mathbb P^1\left(\overline{\mathbb Q}\right)$.

Question

I'm working on the following problem for several days without finding any solution:

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Let $X$ be a complex smooth projective curve and suppose moreover that $X$ is defined over $\overline{\mathbb Q}$. Now consider a finite map $f:X\longrightarrow\mathbb P^1(\mathbb C)$ of degree $d$ and defined over $\overline{\mathbb Q}$.

Question: I don't understand why the branch points of $f$ (if they exist) must lie in $\mathbb P^1\left(\overline{\mathbb Q}\right)\subseteq\mathbb P^1(\mathbb C)$. In particular I'd like to see why one requires that both the map $f$ and the curve $X$ must be defined over $\overline{\mathbb Q}$.

Edit (Why the bounty): I don't understand all the details of the given answer.

Answer 1

Since everything is defined over $\overline{\mathbb{Q}}$, $f$ is fixed by any automorphism of an extension of $\overline{\mathbb{Q}}$ which acts as the identity on $\overline{\mathbb{Q}}$. Since every such automorphism preserves the set of branch points of $f$, and this set is finite, it follows that the branch points must be in $\mathbb{P}^1(\overline{\mathbb{Q}})$.