Hi Robert,
thanks for your answer!
Just some thoughts of mine which might not be "thought to the end":
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 :-)
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.
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
:-)))
5.
Design Question:
In Sage, what should generally the roots of unity be ("I" is just one
of them)?
Elements of the "Symbolic Ring"?
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.
Cheers,
gsw
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