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I implemented the algorithm that Felipe Voloch's suggested in his reply to the question:

Subfields of a function field

the algorithm is here:

Subfields of a function field

I considered the function field $F/k$ ($k$ of positive characteristic) defined by following genus 4 curve:

Y^4 +(2*x^7+ 4*x + 4)*Y^2+ x^14

When I ask Magma to list all subfields between $k(x)$ and $k(x,Y)$, Magma gives me:

F1: Y^2 + (2*x^7 + 4*x + 4)*Y + x^14 Genus 2

F2: Y^2 + 4*x^15 + 4*x^14  Genus 0

F3: Y^2 + 4*x^21 + 4*x^15 + 4*x^14 Gesus 2 

I used Felipe's algorithm in this way: I generated all two dimensional subspaces of the homomorphic differentials of $F$, for each of these subspaces let $[v_1, v_2]$ be a basis, I looked at $k(v_2/v_1)$ (according to the algorithm). What I observed was that all rational subfields of the form $k(v_2/v_1)$ are subfields of either of $F1, F2$ or $F3$.

So my question: while there are infinite rational subfields of $F$ which are not contained in $F1, F2$ or $F3$ why all of $k(v_2/v_1)$ are subfield of these subfields?

So basically I'm asking two questions:

  1. What is the characteristics of the subfields of a function field $F$ which contains all $k(v_2/v_1)$ rational subfields such that $v_2, v_1$ are linearly independent homomorphic differentials?

  2. A mathematical proof that explain this phenomenon.

I checked the above observation for few different $k = \mathbb{F}_{5}, \mathbb{F}_7$ and 11 and I got the same result.

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I don't see why changing the base field should change that fact. –  Will Sawin Oct 30 '11 at 17:23
@Will-sawin, Me neither. I was desperate so I just tried different silly stuff, so hopefully something illuminating might happen. Later, I found out the reason of that phenomena: it's because [F:F2] = 2 and hence F is hyperelliptic, so we have that all rational sub field of F up to some degree are all in F2. then deg(v2/v1) <= |(v1)| 2g-2. and hence v2/v1 in F2. But still my question is that if there's anything special about those rational fields or we are just generating random rational function fields, in that case, the question is that what's the attraction of Felipe algorithm then? –  Syed Nov 14 '11 at 22:06

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