There is a (sort of new) fast probabilistic algorithm for computing Galois groups, due to Nikolai Durov:
- N. V. Durov, Computation of the Galois group of a polynomial with rational coefficients. I. (Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 319 (2004), Vopr.Teor. Predst. Algebr. i Grupp. 11, 117–198, 301; English translation in J. Math. Sci. (N. Y.) 134 (2006), no. 6, 2511–2548 (MR2006b:12006) - N. V. Durov, Computation of the Galois group of a polynomial with rational coefficients. II. (Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 321 (2005), Vopr. Teor. Predst. Algebr. i Grupp. 12, 90–135, 298; English translation in J. Math. Sci. (N. Y.) 136 (2006), no. 3, 3880–3907 (MR2006e:12004) It might be a good project to implement that in sage. Cheers, J On Wednesday, July 25, 2012 8:25:42 AM UTC+1, Nils Bruin wrote: > > On Jul 25, 12:10 am, David Loeffler <[email protected]> > wrote: > > The hard question is to compute a defining > > polynomial for the splitting field of f, identify all the roots of f > > as elements of that field, and determine each element of the Galois > > group in terms of how it acts on a single generator of the splitting > > field. > > That's one way of realizing the galois group "concretely". However, > the normal way would be to determine *a* splitting field, as a > subfield of C or Qp-bar for some good (probably preferably highly > split) prime p and present the Galois action as a permutation on these > (approximate) roots. That's not nearly as expensive as computing a > splitting field as an algebraic extension. In fact, it's mostly a side- > effect of the computations you have to do to determine the galois > group "abstractly" to begin with. > -- -- To post to this group, send an email to [email protected] To unsubscribe from this group, send an email to [email protected] For more options, visit this group at http://groups.google.com/group/sage-devel URL: http://www.sagemath.org
