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][1] (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 $\mathbb{C}$, 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?


  [1]: http://math.stanford.edu/~akshay/research/BookFF.pdf