Restrictions of $Gal(\mathbb{C}/\mathbb{Q})$ to a finite extension $\mathbb{K}$ of $\mathbb{Q}$

Question

I have a doubt. Let $\mathbb{K}$ be a finite extension of $\mathbb{Q}$. May I infer that the number of distinct restrictions of $Gal(\mathbb{C}/\mathbb{Q})$ to $\mathbb{K}$ must be finite? If yes, why?

Answer 1

I hope I am not mistaking the comments above, if I got it right, the question is: Given some $\varphi\in\mathrm{Aut}_{\mathbb{Q}}(\mathbb{K})$, are there inifinitely many $\psi\in\mathrm{Aut}(\mathbb{C})$ with $\psi\vert_{\mathbb{K}}=\varphi$?

The answer to this is yes, for example theorem 7 of this paper (link is taken from this question) states:

Every automorphism of a subfield of $\mathbb{C}$ may be extended to an automorphism of $\mathbb{C}$.

So, as $\mathbb{K}$ is finite over $\mathbb{Q}$, you might choose inifinitely many algebraically independent (over $\mathbb{K}$) elements $\alpha\in\mathbb{C}\setminus\mathbb{K}$, with minimal polynomials, say, $f_\alpha\in\mathbb{K}[x]$, and independently set each of them to arbitrary roots of the $\varphi(f_\alpha)$, to get inifinetly many different extensions to $\mathrm{Aut}(\mathbb{C})$ (which by construction restrict to $\varphi$ on $\mathbb{K}$).