On 2012-11-21, P Purkayastha <ppu...@gmail.com> wrote:
> On 11/21/2012 10:53 PM, Simon King wrote:
>> Hi all!
>>
>> On 2012-11-21, P Purkayastha <ppu...@gmail.com> wrote:
>>> In fact, the behavior in Sage is *inconsistent*, and I think in this
>>> particular case, the inconsistency should get priority of getting fixed
>>> over trying to enforce rigor.:
>>
>> I wouldn't think it is a fix, *unless* it is done in a rigorous way.
>>
>> What's the problem here? We can devide the elements of one integral domain
>> R1 by some elements of another integral domain R2, but (for good reasons!)
>> we can't multiply elements of R1 with elements of the fraction field of R2.
>>
>> A potential solution (and it could be that this is what you suggested)
>> is to say: If there is a coercion map phi of R2 into R1, then Frac(R2)
>> acts on R1 by multiplication, via (p/q)*r1 = phi(p)*r1/phi(q).
>
> What I am asking for is quite simple. If I see a fractional element f =
> p/q during multiplication or division (like p/q*alpha) then in the
> appropriate operation, I would simply run
>
> self(f.numerator())/self(f.denominator())*alpha.
>
> Probably there are more efficient ways of doing this. This brings me to
> the last problem:
>
> GF(5)(3) + 2/3 or 2/3 + GF(5)(3), both result in errors. If we allow
> multiplication, then again there is an inconsistency with addition and
> subtraction.
no, this is OK. The thing is that the multiplication is the action of
Z_{(5)} on Z_{(5)}/5Z_{(5)}=GF(5), i.e. GF(5) is an Z_{(5)}-module.
That's all, there is no addition correctly defined for an element of a
ring R and an element of an R-module.
Dima
> Any ideas what should be done here? Is it always correct to
> coerce to the finite field?
>
>
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