On Sep 26, 2008, at 12:48 AM, Georg S. Weber wrote:
> Hi Robert,
>
> thanks for your answer!
>
> Just some thoughts of mine which might not be "thought to the end":
Thanks for bringing this up. The constant I being in SR has annoyed
me too.
> 1.
> Having fixed "I" in Z / pZ, we have it in Qp, via Teichm"uller lift
> (and vice versa via natural reduction).
>
> 2.
> There is (for p being congruent to 1 mod 4) exactly one root "I" of
> the two roots of X^2+1 in Z /pZ,
> who has a representative in ZZ lying in the interval between 1 and p/
> 2.
> (The other root's representative is the negative of that, or after
> addition of p, in between p/2 and p)
> That would be the natural candidate to choose --- and to document :-)
Yep, we could teach Z/pZ and Qp about this specific case. I don't
know if Integers(17)(10) + I should work, but Integers(17)(I) should.
Could/should we do it in such a way that it generalizes (for
cyclotomic or general number fields)? What about p != 1 mod 4, should
the result be in GF(p^2)?
> 3.
> Many of the mathematical papers I work with have a sentence at the
> beginning like
> "Once and for all, and for all p, we fix embeddings of bar{Q_p} into
> CC ... and use that without further notice"
> or even this sentence is dropped, but silently assumed nevertheless.
You mean C_p I assume? Or an embedding bar{Q} -> CC.
> What would suffice is a fixing of embeddings of the algebraic numbers
> into all local fields.
> And often even only of the maximal abelian extension of Q (which, by
> the Kronecker-Weber theorem,
> is exactly the field extension of Q generated by all roots of unity).
> Of course this goes deeply into class field theory (essentially, "is"
> classical class field theory).
>
> 4.
> In practice, working e.g. in the area of the Birch and Swinnerton-Dyer
> conjecture, one "only"
> needs fixed embeddings of some "small" ("manageable") cyclotomic
> extension of Q into CC, and the
> appropriate extension of Qp, for one p. Of course these can be
> constructed "by hand" each and
> every time, but it would be very nice if my computer would do as much
> work for me as possible
> :-)))
All cyclotomic fields will come with an embedding into CC. There is a
natural choice and this will allow arithmetic between them. As I
mentioned before, there are technical difficulties with providing an
embedding into both CC and all Qp, but all Qp could be informed about
how to accept elements from certain number fields (again, there is a
natural choice for I, but less so for other number fields).
> 5.
> Design Question:
> In Sage, what should generally the roots of unity be ("I" is just one
> of them)?
> Elements of the "Symbolic Ring"?
They should be elements of Z[zeta_n].
> 6.
> It would be fine for me if I could turn on a switch "Automatic
> coercion of all algebraic numbers involved"
> and then compute happily in different rings and fields, without
> worrying.
> Even if this would mean that even the "normal" calculations in one and
> the same ring/field take much more time.
Sounds like you want to use QQbar. However, I don't know if you could
do it "without worrying." What is sqrt(-2) * sqrt(-3) - sqrt(6)? More
complicated algebraic numbers could have more subtle relations (or
non-relations, depending on the embedding chosen, which is less than
obvious what the "right" choice is).
For Z/pZ or Zp, The simple case of choosing a root represented in [0,
p/2] is not entirely "natural" For example, the map phi: Q[sqrt(2),
sqrt(3)] -> GF(23^2) as with this choice is *not* a homomorphism, as
11 = phi(sqrt(6)) != phi(sqrt(2)) * phi(sqrt(3)) = 12. A consistent
choice could be made for cyclotomic fields (based on the
factorization of the order of zeta) so the maximal abelian extension
could come with a consistant embedding into all local fields (albeit
with some work, the smallest cyclotomic extension containing Q[sqrt
(2), sqrt(3)] is Q[zeta_24], and it gets much worse than that very
quickly).
What will certainly be supported is the ability to easily specify an
embedding when making a number field. Arithmetic can be done exactly
and consistently between any two number fields with an embedding into
a common field.
- Robert
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