#12269: coercion and conversion for absolute_field
-----------------------------+----------------------------------------------
Reporter: mstreng | Owner: davidloeffler
Type: enhancement | Status: new
Priority: major | Milestone: sage-5.0
Component: number fields | Keywords: coercion conversion
absolute_field number field structure relative absolute
Work_issues: | Upstream: N/A
Reviewer: | Author:
Merged: | Dependencies:
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Old description:
> {{{
> sage: x = var('x')
> sage: K1.<a1> = CyclotomicField(11)
> sage: K2.<a2> = K1.extension(x^2 - 3)
> sage: K3.<a3> = K2.extension(x^2 + 1) # Let's make a big relative number
> field
> sage: t=a1+6*a2+a3*a1 # with a complicated element.
> sage: L = K3.absolute_field('b')
> sage: L(t)
> # The code at #11869 takes 12 seconds to compute roots of a
> # polynomial, and then doesn't use them.
> # TypeError: No compatible natural embeddings found for Number Field in b
> with .... and Complex Lazy Field
> sage: L.structure() # However, the correct conversion
> map is available and works almost instantly
> #(Isomorphism map:... , Isomorphism map: ...)
> sage: L.structure()[1](t)
> # big output
> sage: L.gen() + t # It would be good if one of the
> structure maps is a coercion, but they aren't at the moment
> # TypeError: unsupported operand parent(s) for '+': ....
> sage: K3(L.gen()) # There are similar problems in the
> other direction
> # TypeError: Cannot coerce element into this number field
> sage: L.structure()[0](L.gen()) # for which the structure maps also
> work.
> # a3 - a2 + a1
> }}}
>
> So to do:
>
> * make both structure maps into conversions (this ticket)
> * make one of the structure maps into a coercion (this ticket)
> * move the "compatible embedding" code of #11869 to the beginning of the
> method, to avoid unnecessary root-finding (#12270)
New description:
{{{
sage: x = var('x')
sage: K1.<a1> = CyclotomicField(11)
sage: K2.<a2> = K1.extension(x^2 - 3)
sage: K3.<a3> = K2.extension(x^2 + 1) # Let's make a big relative number
field
sage: t=a1+6*a2+a3*a1 # with a complicated element.
sage: L = K3.absolute_field('b')
sage: L(t)
# The code at #11869 takes 12 seconds to compute roots of a
# polynomial, and then doesn't use them.
# TypeError: No compatible natural embeddings found for Number Field in b
with .... and Complex Lazy Field
sage: L.structure() # However, the correct conversion
map is available and works almost instantly
#(Isomorphism map:... , Isomorphism map: ...)
sage: L.structure()[1](t)
# big output
sage: L.gen() + t # It would be good if one of the
structure maps is a coercion, but they aren't at the moment
# TypeError: unsupported operand parent(s) for '+': ....
sage: K3(L.gen()) # There are similar problems in the
other direction
# TypeError: Cannot coerce element into this number field
sage: L.structure()[0](L.gen()) # for which the structure maps also
work.
# a3 - a2 + a1
}}}
So this ticket is meant to:
* make both structure maps into conversions
* maybe make one or both of the structure maps into a coercion
Speeding up #11869 is not really related and is #12270.
--
Comment(by mstreng):
Replying to [comment:7 mstreng]:
> {{{structure()[1]}}} cannot be a coercion, because it would lead to non-
commuting diagrams of coercions.
Actually, maybe it is the coercion between O to P that shouldn't be there!
All of the following maps are natural
{{{ O <--> M <-- K <--> L --> N <--> P }}}
and a map between O and P sending generator to generator does not commute
with the abovementioned natural maps.
Some more about O and P:
{{{
sage: O
Number Field in e with defining polynomial x^4 - 10*x^2 + 1
sage: P
Number Field in e with defining polynomial x^4 - 10*x^2 + 1
sage: O is P
False
sage: O == P
True
sage: RR(O.structure()[1].domain().base_field().gen())
-1.41421356237309
sage: RR(P.structure()[1].domain().base_field().gen())
1.41421356237309
}}}
O and P are really different, I don't understand True for "==" here. Is
that a bug?
{{{
sage: K is L
False
sage: K == L
False
}}}
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/12269#comment:10>
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