On 23/01/2008, Martin Albrecht <[EMAIL PROTECTED]> wrote: > > > > By contrast F.multiplicative_gen() does make sense for all finite > > > fields so should be provided, though not necessarily computed until > > > requested for the reasons given by Martin. (It seems that with the > > > current implementation of non-prime fiinite fields this comes for > > > free, but that might change.) > > > > Only for the Givaro ones, i.e., up to 2^16. For general fields it is not > > for free, unfortunately (I think). > > I think it is 'free' for all extension fields because we either represent the > field elements as powers of the generators or as polynomials in the > generator.
I don't see this. If GF(p^n) is represented as GF(p)[X]/(f(X)) where f(X) is an arbitrary degree n irreducible in GF(p)[X], then finding the multiplicative generator will take some work even though that generator will, of course, be represented as a polynomial in the generator a (= root of f). I am not familiar with the best algorithms for finding the (multiplicative) generator for large fields, but I'm sure they are well documented. Of course a standard baby-step-giant-step would work if p^n-1 was small enough (to be factored, for a start). John > > Martin > > > -- > name: Martin Albrecht > _pgp: http://pgp.mit.edu:11371/pks/lookup?op=get&search=0x8EF0DC99 > _www: http://www.informatik.uni-bremen.de/~malb > _jab: [EMAIL PROTECTED] > > > > > -- John Cremona --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-devel@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
