Consider the field extension $L\subseteq K\subseteq \mathbb C$ where $K/L$ is finite. I must show that the set $\{\sigma_{|K}\,:\,\sigma\in\operatorname{Gal}( \mathbb C/L)\}$ is finite, but I have some problems.
Certainly I know that $|\operatorname{Gal}( K/L)|<\infty$ but it seems that this fact doesn't help, infact if $\sigma\in\operatorname{Gal}( \mathbb C/L)$, then $\sigma(K)\subseteq\overline L$, so I can't conclude that $\sigma_{|K}\in\operatorname{Gal}( K/L)$.
Any suggestion?
If $K/L$ is finite, let us say $\lvert K : L \rvert = n$, then every element of $K$ is algebraic over $L$, of degree at most $n$, that is, every element of $K$ is a root of a (nonzero) polynomial in $L[x]$, of degree at most $n$.
Let $a_{1}, a_{2}, \dots, a_{n}$ be a basis of $K$ as a vector space over $L$. Since $\sigma$ fixes $L$ elementwise, $\sigma\mid_{K}$ is determined by the images $\sigma\mid_{K}(a_{i})$ of the $a_{i}$.
Again since $\sigma$ fixes $L$ elementwise, each $a_{i}$ can only be sent by $\sigma$ to a root of its minimal polynomial over $L$, which by the above has degree at most $n$.
All in all, you have at most $n^{n}$ possibilities.