Field Extension problem beyond $\mathbb C$ 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.  
 A: The field of meromorphic functions on $\mathbb{C}$ is huge so I don't expect that this question has a reasonable general answer. One might ask about the fields between $\mathbb{C}$ and meromorphic functions on the Riemann sphere; these are just the rational functions $\mathbb{C}(x)$.
Since $\mathbb{C}$ is already algebraically closed, any nontrivial field between $\mathbb{C}$ and $\mathbb{C}(x)$ necessarily has transcendence degree $1$. Such a field $F$ necessarily lies between $\mathbb{C}(x)$ and $\mathbb{C}(f)$ for some $f \in \mathbb{C}(x)$. $\mathbb{C}(x)$ is always a finite extension of $\mathbb{C}(f)$ (exercise), so the inclusion $F \to \mathbb{C}(x)$ corresponds in the standard way to a branched cover of compact Riemann surfaces (equivalently, smooth projective algebraic curves over $\mathbb{C}$)
$$\mathbb{CP}^1 \to S$$
where $S$ is the Riemann surface with function field $F$. By Riemann-Hurwitz, this can only occur if $S \cong \mathbb{CP}^1$, hence we can choose $f$ so that $F \cong \mathbb{C}(f)$. Thus all nontrivial subfields of $\mathbb{C}(x)$ are of the form $\mathbb{C}(f)$ for some rational function $f$. 
On the other hand, $\text{Aut}_{\mathbb{C}}(\mathbb{C}(x))$ is an interesting group; explicitly it consists of all Möbius transformations $z \mapsto \frac{az + b}{cz + d}, ad - bc \neq 0$ and abstractly it is $\text{PGL}_2(\mathbb{C}) \cong \text{PSL}_2(\mathbb{C})$, the projective special linear group in $2$ dimensions over $\mathbb{C}$. The fixed field of any subgroup of $\text{PGL}_2(\mathbb{C})$ is therefore a subfield of $\mathbb{C}(x)$. Special among these are the finite subgroups. By an averaging argument each of these are contained in the projective special unitary group $\text{PSU}_2$, which is well-known to be isomorphic to $\text{SO}(3)$, and the finite subgroups of this are (more or less) the groups of symmetries of the Platonic solids. The study of the finite subgroups of any of the above related groups is quite fascinating; one entry into further study is the various answers in this MO thread. 
