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I'm asking this question as a follow up to the Felipe Voloch's answer to this question:

Subfields of a function field

which you can read it here:

Subfields of a function field

(I just didn't have much faith in attracting attention to an already-answered question, so I started a new one).

So, my question is that I have no idea why the mentioned algorithm works? Basically how one proves that every subfield of $F$ corresponds to a subspace of the space of Global homomorphic differentials of $F$. Unfortunately, the "(proof left to the reader)", didn't worked out for me.

I would be more than grateful, if you suggest me a reference that helps me to tackles the math behind the algorithm.

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    $\begingroup$ Not every subspace correspond to a (proper) subfield. But you get a subspace from a subfield by looking at the pullbacks of the holomorphic differentials of the subfield. I was just using the fact that, for a non-hyperelliptic curve, the canonical morphism is an embedding. This is all in Hartshorne. $\endgroup$
    – Felipe Voloch
    Commented Aug 10, 2011 at 23:40

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