Hi Nathann,
On 2015-09-18, Nathann Cohen <nathann.co...@gmail.com> wrote:
>>> sage: K1 = GF(8,'x')
>>> sage: K2 = GF(8,'y')
>>> sage: K1(1) == K2(1)
>>> False
>>
>> Not a bug, because cross-parent comparison would only make sense if
>> there is some kind of a canonical map of one parent (and not just of a
>> single element!) into the other or of both parents into a third one.
>
> What I think is unreliable is that "deep down" you decide that two finite
> fields are equal (or not) based on a string comparison.
>
> If two functions create independently two GF(8,'x') then you decide that
> they mean the same, and if two functions create a GF(8,'y') and GF(8,'x')
> then you decide that they mean something different.
What do you mean by "mean"?
Certainly the two fields are isomorphic. But that's not the point here.
So, let me explain what I mean by "mean".
The question is how we can find a system C ("canonical" resp. "coercion")
of homomorphisms between parents in Sage that contains identity morphisms
and is closed under composition and such that for each pair of parents
there is at most one homomorphism in C having the given pair of parents
as domain and codomain.
If there is an isomorphism in C between parents P1 and P2 then there is
a canonical way to identify elements of P1 with elements of P2. But if
there is not, then Sage developers found it difficult to admit that
elements of P1 have *anything* to do with elements of P2.
Thus, if there is no canonical morphism between the parents P1, P2, then
their elements should always evaluate unequal --- because they live in
separate worlds.
The automorphism group of GF(p^n) for n>1 is non-trivial. Thus, it is
not straight forward to single out a "canonical" isomorphism between two
non-identic copies of GF(p^n). In that sense, they do not mean the same.
So, that's what *I* mean by "mean". If we build a system C of canonical
morphisms (aka "coercions") and there is no isomorphism in C with domain
GF(8,'y') and codomain GF(8,'x') then GF(8,'y') and GF(8,'x') mean
something different. Otherwise, they do *not* mean "the same", but at
least one can make them "the same" in a canonical way.
> If would feel better if two GF(8,'x') were always different, or if all
> GF(8,<whatever>) were equal.
There are good reasons to say that if you construct GF(8,'x') twice then
the results are not only the same but *identical*. Even better reasons
are imply that GF(8,'x') and GF(8,'y') can not be identical.
One can of course try the following rule for constructing C: If the
minimal polynomial used in the construction of GF(8,'x') is obtained
from the minimal polynomial used in the construction of GF(8,'y') by
substitution y->x, then the map extends to an isomorphism that shall
belong to C. If the minimal polynomials are not mapped in that way, then
*no* isomorphism between the two fields should belong to C.
I simply don't know if such a rule gives rise to an inconsistency in the
"global picture" (i.e., in the system of *all* coercions in Sage). But
the rule does seem reasonable to me.
Best regards,
Simon
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