Ideally I should be the element x in ZZ[x]/(x^2+1) (and not QQ[x]/
(x^2+1)).
Likewise zeta[n] could be the element y in ZZ[y]/(y^n-1).
These rings should have automatic coercions to many other rings
(including
of course the symbolic ring).
Michel
On Jun 12, 8:04 pm, Robert Bradshaw <[EMAIL PROTECTED]>
wrote:
> On Jun 12, 2007, at 8:21 AM, Nick Alexander wrote:
>
>
>
> > "William Stein" <[EMAIL PROTECTED]> writes:
>
> >> On 6/12/07, Robert Bradshaw <[EMAIL PROTECTED]> wrote:
> >>> I think the default I should belong to the number field Q[I] (or
> >>> perhaps even the ring of integers) to start with (together with a
> >>> fixed embedding into C). It would be coerced into C, the symbolic
> >>> ring, etc. as needed.
>
> >> With the current coercion and number field model, I very strongly
> >> disagree -- the I should be symbolic. Anyway,
> >> you didn't give any argument for why you think "I" should be a
> >> number field
> >> element instead of a symbolic element. I would probably agree
> >> with you if you
> >> were to give a good argument for why you think that, and if number
> >> fields were
> >> much more sophisticated, and if the coercion model were suitably
> >> enhanced.
>
> > Making I different from any other symbol is confusing to me, too. Are
> > we going to declare zeta2, ..., zetan to be the other roots of unity?
> > And have the all special cased too? The symbolic functionality is at
> > least symmetric.
>
> My reasoning is the same as for making sqrt(2), etc. belong to a
> number field rather than the symbolics. First is simply speed,
> especially over compound objects--e.g. a power series over the
> symbolic ring is really slow. I also think it's unnatural to have to
> manually simplify "constant" expressions like 1/(1+I). Symmetry is
> already broken, as I is already special
>
> sage: I^2
> -1
>
> Maple and Mathematica have this behavior:
>
> sage: maple("1/(1+I)")
> 1/2-1/2*I
>
> sage: mathematica("1/(1+I)")
> 1 I
> - - -
> 2 2
>
> Doing this (in general), symbolically, would be difficult. Instead,
> the arithmetic operations will just keep "piling up." If one wants
> this behavior one could explicitly coerce into the symbolic field.
> Also, I think things should start out in as restricted a ring as
> possible, both to have coercion work and for specific functionality,
> e.g. one should be able to do
>
> sage: (1+I/3).charpoly('x')
> x^2 - 2*x + 10/9
>
> I am assuming an enhanced coercion model and number fields with
> embeddings (hopefully in the near future).
>
> -Robert
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