Let $K$ be an infinite field, if $K(X)$ is the field of rational function I want to find the Galois group of the extension $K(X)/K$.
Lemma 1: If $L$ is a field such that $K\subsetneq L\subseteq K(X)$ then $[K(X):L]$ is finite.
Proof. It is easy to show that $X$ is algebraic over $L$, so $K(X)/L$ is a finite extension.
Lemma 2: $Gal(K(X)/K)$ contains only finite (proper) subgroups.
proof. Suppose that $H<G$ is infinite; for the lemma 1 we have that $[K(X): Fix(H)]=n$, and so $|Gal(K(X)/Fix(H))|\le n$. But $Gal(K(X)/Fix(H))\supseteq H$ and this is a contraddiction.
Now I know that the group $Gal (K(X)/K)$ is a group with only finite subgroups, but I can't find other informations about its structure. Maybe this group depends strongly from the field $K$.
Thanks in advance.