#14990: Implement algebraic closures of finite fields
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Reporter: pbruin | Owner:
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-6.3
Component: algebra | Resolution:
Keywords: finite field | Merged in:
algebraic closure | Reviewers:
Authors: Peter Bruin | Work issues:
Report Upstream: N/A | Commit:
Branch: u/pbruin/14990 | b1d4424ec179f4975a54a14993e4c842b3fb39f5
Dependencies: #14958, #13214 | Stopgaps:
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Comment (by vdelecroix):
Dear Peter,
I like very much the documentation in `algebraic_closure`. Thank you.
For each problem below, I think we need to find a solution which might be:
either write specifications in the documentation or provide a fix that
change the behaviour.
1) Equality among algebraic closures is not constant over time:
{{{
sage: F1 = AlgebraicClosureFiniteField(GF(3), 'z', use_database=False)
sage: F2 = AlgebraicClosureFiniteField(GF(3), 'z', use_database=False)
sage: F1 == F2
True
sage: F1.gen(3)
z3
sage: F1 == F2
False
sage: F2.gen(3)
z3
sage: F1 == F2
True
}}}
It becomes really weird when we play with pickling
{{{
sage: p = 100003
sage: K = GF(p).algebraic_closure()
sage: K2 = loads(dumps(K))
sage: K.gen(3)
z3
sage: K == K2
False
}}}
One fix is to use identity in comparisons of the parent themselves as I
suggested in previous comments. The only drawback I see is that `K ==
loads(dumps(K))` will be `False`. I think it would be safer that way.
2) Get a very strange error from conversions between different algebraic
closures that can be fixed introducing more type checking in the methods.
{{{
sage: from sage.rings.algebraic_closure_finite_field import
AlgebraicClosureFiniteField
sage: F1 = AlgebraicClosureFiniteField(GF(3), 'z')
sage: F2 = AlgebraicClosureFiniteField(GF(3), 'z')
sage: F1(F2.gen(1))
Traceback (most recent call last):
...
TypeError: no canonical coercion from Algebraic closure of Finite Field of
size 3 to Finite Field of size 3
}}}
3) (minor) The cache in `algebraic_closure` does not take into account the
default arguments.
{{{
sage: K1 = GF(5).algebraic_closure(implementation='pseudo_conway',
use_database=True)
sage: K2 = GF(5).algebraic_closure(implementation='pseudo_conway')
sage: K4 = GF(5).algebraic_closure()
sage: K1 is K2 or K1 is K3 or K2 is K3
False
}}}
One fix is to move the cache at the level of
`AlgebraicClosureFiniteField`, but on the other hand it is good to have
the cache at a Cython level.
That's all
Vincent
--
Ticket URL: <http://trac.sagemath.org/ticket/14990#comment:85>
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