#6046: [with patch, needs review] Implement local and global heights for number
field elements
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Reporter: cremona | Owner: was
Type: enhancement | Status: new
Priority: major | Milestone: sage-4.0.1
Component: number theory | Keywords: number field height
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Comment(by cremona):
Thanks for your comments, Francis, and for fixing this up to work with
relative extensions.
Your comments and fixes for numerator/denominator ideals are *wrong*!
number field elements have numerators and denominators which are dependent
on the basis used to represent them, and are not what I meant or need --
as I thought my doctest made clear!
We must sort out this mess with the embeddings. Your first doctest
failure (with different behaviour in different runs) must mean that the
ordering of the embeddings is not deterministic. That must be fixed, say
by ordering the roots when the embeddings are found. Secondly we must
have a fool-proof way of determining which embeddings are real when you do
K.complex_embeddings(). What I would prefer is to have a function called
(perhaps) K.archimedean_completions() which returns a list of r+s
embeddings, the first r into a RealField and the last s into a
ComplexField (or even a list of two lists of embeddings, of lengths r and
s respectively). That way the codomain of a real embedding would be a
RealField which could easily be tested for. While we are at it the
complex list would only contain one of each pair.
So, how to implement this? Not problem with the list of real embeddings,
though they should be sorted by the natural real ordering on the image og
K.gen(); for the non-real ones we could first find all embeddings into
CC, then sort them by their imaginary parts and then take the last s of
these (where s could be defined to be (n-r)/2 where n is the degree and r
the number of real embeddings).
Does that sould workable?
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/6046#comment:2>
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