On Jun 12, 2007, at 11:31 AM, Michel wrote:
> Ideally I should be the element x in ZZ[x]/(x^2+1) (and not QQ[x]/
> (x^2+1)).
I agree, but currently we have much better support for number fields
than orders or quotient rings. Hopefully that'll change (r.e. David
Roe's SEP).
> Likewise zeta[n] could be the element y in ZZ[y]/(y^n-1).
Perhaps. Zeta is used for a lot of other things than the n-th roots
of unity though, that would be my only concern.
>
> 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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