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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 ├ętale 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?
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I don't think that PARI has tailor-made commands to compute function field Galois groups. The (non-free) package MAGMA computes Galois groups over $\mathbb{F}_q(t)$ and does function field arithmetic. There are people around here who will know the answer for SAGE, or you could just check up the SAGE documentation online. –  Tony Scholl Aug 2 '10 at 19:16
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You do not need a definition of universal cover to have a notion of fundamental group. You can define the fundamental group of a pointed curve using a Tannakian recipe, as the automorphism group of a certain functor from the category of etale covers to the category of finite sets. –  S. Carnahan Aug 2 '10 at 20:15
    
Oops, that should be either finite etale covers to finite sets or etale covers to sets, but not my particular mixture. –  S. Carnahan Aug 2 '10 at 20:16
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