#16613: fix content of polynomials
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Reporter: vdelecroix | Owner:
Type: defect | Status: needs_info
Priority: major | Milestone: sage-6.4
Component: commutative algebra | Resolution:
Keywords: | Merged in:
Authors: Vincent Delecroix | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
Dependencies: | Stopgaps:
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Comment (by bruno):
I am trying to provide some code for this. I'd like opinions on two
points:
1. To avoid future inconsistencies, I plan to implement not only `content`
and `content_ideal`, but also the related `primitive_part` and
`is_primitive` for all polynomial rings with consistent semantics. Do you
think it is appropriate to do so in this ticket, or do you think that it's
better to stick with `content` and `content_ideal` only?
2. What should it be the semantic for fields?
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==== Remarks for the semantics for fields:
Currently, there is a method `is_primitive` which uses three distinct (and
documented!) semantics:
* An irreducible polynomial of degree ''m'' over **F**,,p,, is
primitive if its roots generate **F**,,p^m,,;
* A polynomial over a ring is primitive if its coefficients generate
the unit ideal;
* Primitivity is undefined for polynomials over an infinite field, that
is an exception is raised.
I guess that there should not be any opposition for `content_ideal` to
return the unit ideal. For `content`, I see three possibilities (there may
be more):
* Raise an exception;
* Return `1` (or more precisely `self._parent.one()`);
* Return `gcd(self.coefficients())` (knowing that `gcd(2/3,4/5)=2/15`
with the current semantics for `gcd`.
And I would say that in all three cases, `primitive_part` should return
`self` divided by its content, using for content the same semantic as
above. Finally, the case of `is_primitive` is complicated because of the
semantic for finite fields. One possibility could be to find another name
for this semantic for finite fields but 1. I guess that "primitive" is
really the word used in mathematics for this property, and 2. this may
break some existing code (and given the current discussions on sage-devel,
I doubt the community would favor this!).
--
Ticket URL: <http://trac.sagemath.org/ticket/16613#comment:6>
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