#14084: Wrong domain of the fraction field construction functor
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Reporter: SimonKing | Owner: roed
Type: defect | Status: needs_review
Priority: major | Milestone: sage-5.7
Component: padics | Resolution:
Keywords: | Work issues:
Report Upstream: N/A | Reviewers:
Authors: Simon King | Merged in:
Dependencies: | Stopgaps:
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Comment (by nthiery):
Replying to [comment:15 nbruin]:
> Replying to [comment:5 nthiery]:
> > The above is in fact alright because Fields is a full subcategory of
> > Rings. So by going from one to the other, one don't change the
> > structure under consideration. One is just learning more properties of
> > this structure.
>
> For the most part I probably agree with that. However, how does this
relate to additional concepts such as "finitely generated"?
>
> The following is probably not entirely kosher in terms of mathematical
categories, but it may affect the colloquial use of them in computer
algebra:
>
> The Gaussian field `QQ(i)` would generally be considered a finitely
generated field, but as a ring it would not be finitely generated (no
characteristic 0 field is). Of course one should specify over what object
they are finitely generated, and the discrepancy comes from the different
default choices for fields and rings: The prime fields `QQ` and `GF(p)`
for fields versus `ZZ` for commutative rings.
>
> I would love to see that we can use the "category" field as a dynamic
attribute as suggested, because it does make these `is_field` tests
wonderfully fast without requiring complicated additional caching. But I
am concerned that phenomena (or misconceptions) as above might come back
and bite us if we're coming to depend on such tricks extensively. Can you
ease that concern?
I guess you pointed the key fact above: one can have some shortcuts
for the users convenience, but for a robust answer one should always
specify for which category the thing is finitely generated. For the
same reason .gens should only be a shortcut. For robust results, one
should always specify the category. Which is my motivation for
promoting the use of semigroup_generators / monoid_generators /
algebra_generators / ... Luckily, in some cases we have nice names
(finite dimensional, basis,...) for those.
Cheers,
Nicolas
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/14084#comment:17>
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