I am posting this problem in order to break the problem in my previous post Field Extension problem beyond $\mathbb C$.
Notation: $M(\mathbb C):=$ Field of all meromorphic functions on $\mathbb C$, $K:=$ a proper subfield between $\mathbb C$ and $M(\mathbb C)$
Question 1: Let $\sigma\in Aut(K)$ and $c$ is a constant function. Is it true $\sigma(c)$ is again a constant function ?
I will post subsequent questions after getting a good answer (either by myself or someone else). Let us try how far we can go!