There are lots of fields between $\mathbb C$ and Meromorphic Functions on $\mathbb C$. For example set of "All Even Meromorphic Functions on $\mathbb C$'' is a subfield between $\mathbb C$ and Meromorphic Functions on $\mathbb C$.
Question: How to categorize such subfields ?
I have no idea whether somebody studied this or not. If somebody give me some reference it will be helpful to me.