#13214: Frobenius endomorphism over finite fields
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Reporter: caruso | Owner: AlexGhitza
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-5.11
Component: basic arithmetic | Resolution:
Keywords: frobenius finite fields | Work issues: does not build
Report Upstream: N/A | Reviewers: Paul Zimmermann
Authors: Xavier Caruso | Merged in:
Dependencies: #13184 | Stopgaps:
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Comment (by caruso):
Replying to [comment:25 pbruin]:
> This looks interesting. One question at the moment: it does not appear
to relate to the basic functionality that currently exists (in
`sage/rings/finite_rings/homset.py`). Is that intentional?
Yes, more or less.
Actually, my first motivation was to implement Frobenius endomorphism and,
according to me, FrobeniusEndomorphism should not derive from
RingHomomorphism_im_gens. Indeed, defining Frobenius endomorphisms by
giving its value on the generator is not really appropriate for many
operations we would like to perform (i.e. evaluation if the field has a
small characteristic and a big size, composition, computation of the order
or of the subfield of fixed points).
On the other hand, I agree that it makes sense to derive the various
classes of embeddings from FiniteFieldHomomorphism_im_gens. I can actually
change this if you think that it is better.
> As for the differing purposes of the two approaches, there should
probably be two categories into which a finite field can be put:
>
> - the category of all finite fields. In this category, between any two
objects there are either several morphisms or none at all, but no
canonical one.
>
> - the category of finite subfields of a given algebraic closure of
'''F''',,''p'',,. In this category there is at most one morphism beteen
any two objects, namely the inclusion qua subfields of the given algebraic
closure.
This sounds good.
And the second category should be a subcategory of the first one, right (a
finite subfield of a given algebraic closure of '''F''',,''p'',, is in
particular a finite field)? If so, all category-independant functions
should go to the first category (even if the second one is the default, as
suggested by Jean-Pierre). This is the case in particular for Frobenius
endomorphisms.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/13214#comment:29>
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