Hi, It is probably a bias of the choice of (additive) generators for finite field extensions which results in the primitive field element also being a generator for the multiplicative group (confusingly called a "primitive element of the finite field").
It is not possible to set GF(q).gen() to always be a generator for the multiplicative group, since it is not always computationally feasible to determine it. In particular requires a factorization of q-1. The finite field constructor for non-prime fields would then also have to be changed to ONLY use primitive elements (since GF(q).gen() must certainly return the generator with respect the the defining polynomial). This would conflict with the desire to allow user-chosen polynomials or polynomials of SAGE's choice which are selected for sparseness (hence speed) rather than as generator of the multiplicative group. Moreover, any such definition for convenience of the user would have to be turned off for q > [some arbitrary bound] and could not be reliable. Thus I don't see how it would be possible to make the definition that GF(q).gen() is a generator for the multiplicative group. For small finite fields, one could set GF(q).gen() to be such a generator. This could either be convenient, or confuse users into thinking that this is more generally True. --David P.S. On the other hand, I find (additive) order confusing and there should be some checking of the index for FF.i below (i.e. give an error if i ! = 0). sage: FF.<x> = GF(7^100); sage: x.order() 7 sage: x x sage: x.multiplicative_order() 323447650962475799134464776910021681085720319890462540093389533139169145963692806000 sage: FF.0 x sage: FF.1 x sage: FF.2 x sage: FF.100 x sage: FF.101 x --~--~---------~--~----~------------~-------~--~----~ 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 -~----------~----~----~----~------~----~------~--~---
