I just realized a source of my confusion. The docstring that I quoted
was not actually wrong in the way that I thought is was, but was
apparently deceptive (to me). Perhaps some people are already aware of
this, but GF(5), GF(25), and GF(5^100) are all different types, and so
have different docstrings, some of which have errors.
sage: F = GF(5)
sage: K = GF(5^2,'x')
sage: L = GF(5^100,'x')
sage: type(F)
<class 'sage.rings.finite_field.FiniteField_prime_modn'>
sage: type(K)
<type 'sage.rings.finite_field_givaro.FiniteField_givaro'>
sage: type(L)
<class 'sage.rings.finite_field_ext_pari.FiniteField_ext_pari'>
I earlier quoted the docstring for
sage.rings.finite_field_givaro.FiniteField_givaro. Assuming that 'small'
means 'implemented using givaro', I guess (from Martin's email) that for
fields F implemented using givaro F.gen() is always a multiplicative
generator. But for prime fields constructed with GF(), F.gen() is always
1, and for large fields, F.gen() is 'just' a root of the defining
polynomial, whatever that happens to be.
In fact, if I force the construction of a givaro-implemented prime
field, then the generator is not 1.
sage: H = sage.rings.finite_field_givaro.FiniteField_givaro(5)
sage: H.gen()
2
So the state of the docstrings is
sage: sage.rings.finite_field.FiniteField_prime_modn.gen?
[...]
Docstring:
Return generator of this finite field.
(Nothing wrong here I suppose, but I think that it is always going to
return 1.)
sage: sage.rings.finite_field_givaro.FiniteField_givaro.gen?
[...]
Docstring:
Return a generator of self. All elements x of self are
expressed as log_{self.gen()}(p) internally. If self is
a prime field this method returns 1.
(The sentence "If self is a prime field..." is wrong, but the first
sentence is correct.)
sage: sage.rings.finite_field_ext_pari.FiniteField_ext_pari.gen?
[...]
Docstring:
Return chosen generator of the finite field. This generator
is a root of the defining polynomial of the finite field, and
is guaranteed to be a generator for the multiplicative group.
(This is wrong because in this case the generator is not guaranteed to
be a generator for the multiplicative group.)
I think that the ultimate source of my confusion is that GF is not a
class, but is actually a function that returns a instance of any one of
at least 3 different classes. I feel that if it's somehow possible I
would much prefer if FiniteField was a type all its own, and if all
finite fields were of type FiniteField, and the difference in
implementations was completely invisible. (I'm pretty sure that there
are other instances of this type of thing in sage.)
At the very least, there should probably be some sort of uniformity in
the docstrings, but I'm not sure exactly how this should work.
(Additionally, I was just thinking that it might be nice to have a
standard "SEE ALSO:" section in the documentation. For example
SEE ALSO: multiplicative_generator()
The online documentation could even automatically generate links for
this.)
Anyway, that was quite a long email. Congratulations and apologies if
you read it all.
-bober
On Tue, 2008-01-22 at 00:16 +0000, Martin Albrecht 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
>
>
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