#11938: Finite fields defined by Conway polynomials: conversion of elements into
subfields
-------------------------------------------------+-------------------------
Reporter: dkrenn | Owner: cpernet
Type: enhancement | Status:
Priority: major | needs_work
Component: finite rings | Milestone:
Keywords: finite field, givaro, Conway | sage-5.11
polynomial, conversion, coercion sd51 | Resolution:
Authors: Daniel Krenn | Merged in:
Report Upstream: N/A | Reviewers: Jean-
Branch: | Pierre Flori
Stopgaps: | Work issues:
| Dependencies: #12084,
| #8335
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Changes (by pbruin):
* status: needs_review => needs_work
Comment:
In the meantime, #8335 changed a lot and this patch does not apply
anymore. On the other hand, after #8335, one can now do
{{{
sage: F9=GF(9,conway=True,prefix='z')
sage: F81=GF(81,conway=True,prefix='z')
sage: F81(F9.gen())
2*z4^3 + 2*z4^2 + 1
}}}
I want to advocate the viewpoint that this feature should only be enabled
when a compatible system of finite fields is expressly requested, either
using the `conway=True` syntax from #8335 or in the future using an
algebraic closure (see #14990).
The fact that elements from `F81` that are actually in `F9` cannot yet be
coerced down into `F9` is not addressed by #8335, though. I guess this
ticket is the place to solve that problem.
It is probably better to put this in `FiniteField_base.pyx`, since the
feature is in principle implementation-independent.
One thing to think about: in the current patch, the down-conversion
involves taking roots in the finite field. This might be the best way for
finite fields that are implemented using Givaro, because it represents
elements as powers of a generator of the multiplicative group. For
general finite fields, it could be more efficient to write down the
inclusion map as an '''F''',,''p'',,-linear map and to find the unique
solution of the corresponding linear system.
--
Ticket URL: <http://trac.sagemath.org/ticket/11938#comment:19>
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