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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.

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    I guess you can start with Galois theory. For one, your example is the fixed field of the automorphism $X \mapsto -X$2012-05-16
  • 1
    Rational functions...2012-05-16

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