I am trying to prove that if $E$ is an infinite field, then the fixed field of $Gal(E(x)/E)$ is $E$.
The first part of the question was to find all automorphisms $x\longmapsto \frac{P(x)}{Q(x)}:E(x)\to E(x),$ which I did. They are of the form $ x \longmapsto \frac{ax+b}{cx+d} $ with $ad-bc \neq 0$.
The second part said to prove that if $E \subset K \subseteq E(x)$ for an intermediate field $K$, then $[E(x):K]$ is finite. I did this.
The third part (which I am stuck on) says to prove that if $E$ is infinite that the fixed field of $Gal(E(x)/E)$ is $E$.
I don't really see how this relates to the previous parts, nor do I know how to finish the proof. Since the fixed field of $Gal(E(x)/E)$ is an intermediate field between $E(x)$ and $E$ I could try to prove that the degree of $E(x)$ over the fixed field of $Gal(E(x)/E)$ is infinite and then apply the second part, but this seems very complicated and I don't know how I would go about it.
Does anyone have any suggestions for me? They would be very much appreciated.
If anyone could elaborate on Pete's hint, I have been trying to figure it out for quite a while now and haven't made any progress. Thank you!