Hi Robert,
On 26 Sep., 10:35, Robert Bradshaw <[EMAIL PROTECTED]>
wrote:
> 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.
Well, it's evil, but I've seen that. It works like this:
Forget all analytic/topological structures, just look at the field
structure.
Then both bar{Q_p} and CC are pure transcendental extension of Qbar,
of
transcendence degree some (infinite) cardinality each, so using the
Axiom
of Choice / Zorns Lemma there is a field injection from bar{Q_p} into
CC.
(The cardinality of the bar{Q_p} transcendence degree being
lesserequal
than that of 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).
>
Hmmm ... and even it had worked for "I", there are plenty of roots of
unity.
Sigh.
> 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.
>
That sounds promising, yeah!
> - Robert
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