I don't like the behavior illustrated below. Briefly, my problem is that
GF(p).gen() gives a generator for the additive group of GF(5), while
GF(p^n).gen() gives a generator for for multiplicative group of GF(p^n)
(n > 1).
I would file this 'complaint' directly as a trac bug report, but the
documentation is somewhat clear that this is what is _supposed_ to
happen - i.e., it says
sage: F.gen?
Type: builtin_function_or_method
Base Class: <type 'builtin_function_or_method'>
String Form: <built-in method gen of
sage.rings.finite_field_givaro.FiniteField_givaro object at 0x9c22324>
Namespace: Interactive
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.
[...]
I also know that there is multiplicative_generator() method which always
does the right thing, but I still don't like this inconsistency.
Anyway, perhaps I'll turn my 'complaint' into a trac ticket, but I won't
bother if others don't consider this to be a bug.
-bober
p.s. The complaint illustrated:
sage: F = GF(25,name='x')
sage: x = F.gen()
sage: x # x generates the multiplicative group
x
sage: [x^i for i in range(25)]
[1,
x,
x + 3,
4*x + 3,
2*x + 2,
4*x + 1,
2,
2*x,
2*x + 1,
3*x + 1,
4*x + 4,
3*x + 2,
4,
4*x,
4*x + 2,
x + 2,
3*x + 3,
x + 4,
3,
3*x,
3*x + 4,
2*x + 4,
x + 1,
2*x + 3,
1]
sage: K = GF(5, name='y')
sage: y # y generates the additive group
1
sage: y * y
1
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