#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):
Replying to [comment:4 fwclarke]:
> Yes; no time to experiment more with this, but I note that `places` in
`number_field.py` seems to do what's needed:
> {{{
> def places(self, all_complex=False, prec=None):
> """
> Return the collection of all places of self. By default, this
> returns the set of real places as homomorphisms into RIF first,
> followed by a choice of one of each pair of complex conjugate
> homomorphisms into CIF.
>
> On the other hand, if prec is not None, we simply return places
> into RealField(prec) and ComplexField(prec) (or RDF, CDF if
> prec=53).
>
> ...
> }}}
> so that
> {{{
> sage: K.<a> = NumberField(x^4+3*x^2-17)
> sage: K.places()
>
> [Ring morphism:
> From: Number Field in a with defining polynomial x^4 + 3*x^2 - 17
> To: Real Field with 106 bits of precision
> Defn: a |--> -1.699259307373674805202512065354,
> Ring morphism:
> From: Number Field in a with defining polynomial x^4 + 3*x^2 - 17
> To: Real Field with 106 bits of precision
> Defn: a |--> 1.699259307373674805202512065354,
> Ring morphism:
> From: Number Field in a with defining polynomial x^4 + 3*x^2 - 17
> To: Complex Field with 53 bits of precision
> Defn: a |--> 2.42641344244876*I]
> }}}
>
> Not defined (yet!) for relative number fields.
>
This does exactly what I was suggesting. So I will amend my heights code
to use places instead of embeddings. I note that places() does not work
for QQ though, and that will have to change!
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/6046#comment:6>
Sage <http://sagemath.org/>
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