Thoughts on this thread:
For finite fields (or any other fields) the concept of additive
generator makes no sense -- only finite prime fields have one and it
is hardly a useful concept then since every nonzero element is one.
It's different if talking about generators (plural!) which I think is
what David Kohel meant, but here we are talking about what F.gen()
should be.
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.)
As for F.gen() it might be worth thinking more generally about what
one means by "*the* generator" of a field. Again the concept only
makes sense for some fields: those which are generated by a single
element over their prime field (so: any number field, any finite
field, or the function fields in one variable over QQ or GF(p)); or,
more generally for fields constructed as relative extensions of other
fields, we could include finite (algebraic) extensions of any field at
all, and function fields in one variable over any field.
Arguably, neither QQ nor GF(p) require any generators at all, but for
consistency with general number fields and finite fields it would be
reasonbly to have F.gen() return any nonzero element, and why not fix
it as F(1).
There should certainly be no expectation that F.gen() be a
multiplicative generator, even if it often is -- so perhaps a warning
in the documentation should tell users to use F.multiplicative_gen()
if that's what they really need.
As for order(), I would be happy for it not to be implemented at all
for elements of a field! For additive order it is completely
redundant since all nonzero elements have the same order depending
only on the characteristic. For multiplicative order it is useful
(for nonzero elements of course), but would it not be better, if you
needed mutiplicative orders, to create the multiplicative group of the
field as an abelian group and ask for the order of an element in that
context?
This links in with something I needed to be able to do: given an
element a of GF(q) qith q a non-prime prime power, to return the
degree of a, meaning the smallest n such that a is in GF(p^n). One
could work this out from the (mutiplicative) order of a, which might
be efficient if GF(q)'s multiplicative generator was already known
together with the factorization of q-1. Or it's the degree of the
minimal polynomial of a over GF(p).
More generally, given a field F which is an extension of some base
field k and an element a of F it would be nice to be able to construct
the subfield k(a) of F together with a suitable embedding into F and
an element in it which maps to a. But this is getting too far afield
;), so I'll shut up.
John
On 22/01/2008, Martin Albrecht <[EMAIL PROTECTED]> wrote:
>
> > In this case, the docstring needs to be corrected, because the statement
> > that "All elements x of self are expressed as log_{self.gen()}(p)
> > internally" is not true, right? (Extrapolating from this sentence and my
> > two examples led me to make my previous statements.) Probably it is true
> > that all elements are expressible as polynomials in x, and perhaps this
> > is also the internal representation.
>
> Hi,
>
> It is not. For small extension fields of order < 2^16 the elements are
> represented using Zech logs internally, so for these finite fields the
> docstring "all elements x of self are expressed as log_{self.gen()}(p)
> internally" is true.
>
> 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
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