#4193: Coercion between relative and absolute number fields
---------------------------+------------------------------------------------
Reporter: davidloeffler | Owner: was
Type: defect | Status: new
Priority: major | Milestone: sage-3.4.2
Component: number theory | Keywords:
---------------------------+------------------------------------------------
Changes (by davidloeffler):
* cc: fwclarke (added)
Old description:
> Is there supposed to be a canonical coercion map between a relative
> number field and its associated absolute field?
>
> At the moment (in 3.1.2) there apparently isn't, and trying to force one
> raises a TypeError:
> {{{
> sage: K1.<a> = NumberField(x^3 - 2)
> sage: R.<y> = PolynomialRing(K1).gen()
> sage: K2.<b> = K1.extension(y^2 - a)
> sage: K2abs = K2.absolute_field('w')
> sage: K2abs(b)
> TypeError: Cannot coerce element into this number field
> }}}
>
> I suppose it's sort of fair enough as there exist multiple K1-linear
> embeddings from K2 into K2abs, but shouldn't the definition of K2abs give
> a distinguished element of this set, sending the generator b of K2 to the
> generator w of K2abs?
>
> This causes problems elsewhere, as the code for relative orders in number
> fields relies on having such a coercion to do basic element-creation and
> membership-testing routines, and these are all broken as a result:
> {{{
> sage: R = K2.order(b)
> sage: b in R
> False
> sage: bb = R.gens()[1] # b by any other name
> sage: bb == b
> True
> sage: bb.parent() is R
> True
> sage: bb in R
> False
> sage: R(bb) # trying to coerce something into its own parent!
> TypeError: Cannot coerce element into this number field
> }}}
>
> I uncovered this last problem first while trying to fix #4190, or more
> precisely while trying to write a doctest for a fix that I'd already
> written. (I have a fix which works for absolute orders and should work
> for relative orders too, but there's no way it can work given the above
> general brokenness.)
>
> David
New description:
Is there supposed to be a canonical coercion map between a relative number
field and its associated absolute field?
At the moment (in 3.1.2) there apparently isn't, and trying to force one
raises a TypeError:
{{{
sage: K1.<a> = NumberField(x^3 - 2)
sage: R.<y> = PolynomialRing(K1)
sage: K2.<b> = K1.extension(y^2 - a)
sage: K2abs = K2.absolute_field('w')
sage: K2abs(b)
TypeError: Cannot coerce element into this number field
}}}
I suppose it's sort of fair enough as there exist multiple K1-linear
embeddings from K2 into K2abs, but shouldn't the definition of K2abs give
a distinguished element of this set, sending the generator b of K2 to the
generator w of K2abs?
This causes problems elsewhere, as the code for relative orders in number
fields relies on having such a coercion to do basic element-creation and
membership-testing routines, and these are all broken as a result:
{{{
sage: R = K2.order(b)
sage: b in R
False
sage: bb = R.gens()[1] # b by any other name
sage: bb == b
True
sage: bb.parent() is R
True
sage: bb in R
False
sage: R(bb) # trying to coerce something into its own parent!
TypeError: Cannot coerce element into this number field
}}}
I uncovered this last problem first while trying to fix #4190, or more
precisely while trying to write a doctest for a fix that I'd already
written. (I have a fix which works for absolute orders and should work for
relative orders too, but there's no way it can work given the above
general brokenness.)
David
--
Comment:
I'm pleased to report that this seems to be fixed by fwclarke's patch for
#5508. There isn't automatic coercion between relative + absolute fields
but it's debatable whether there should be; the more severe issue was the
problems with orders assuming that there was in their coercion code, which
is now fixed. Well done that man.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/4193#comment:1>
Sage <http://sagemath.org/>
Sage - Open Source Mathematical Software: Building the Car Instead of
Reinventing the Wheel
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