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

the algorithm is here:

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:

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?

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.