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

which you can read it here:

(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.