Is there a way to compute Galois groups of function field extensions using Pari or Sage? Given a polynomial, $p(x,t) \in \mathbb{F}_p[t][x]$ we can try to solve for $x(t)$, but if no such polynomial exists we can make the field extension $L = \mathbb{F}_p[t][x]/p(x,t)$. For an irredicible p(x,t), I would like to find the Galois group $\mathrm{Gal}(L / \mathbb{F}_p[t])$.

I've read (see p. 3) these field extensions correspond to branched covers of the projective plane $\mathbb{P}^1$ over $\mathbb{F}_p$. If we did this over $\mathbb{C}$, the Galois group is the group of deck-transformations of this $\mathbb{P}^1$-cover. But since we're working over a finite field, this paves the way for the étale fundamental group (once we have a definition of universal cover... which I suspect was given by Grothendieck).

So two main questions,

- Does software exist to help me add and multiply elements of these field extensions? find their Galois group?
- Does this example have an interpretation in terms of Etale topology?