#8335: Finite field lattices via Conway polynomials
-------------------------------------+-------------------------------------
Reporter: roed | Owner: AlexGhitza
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-5.11
Component: algebra | Resolution:
Keywords: days49 sd51 | Merged in:
Authors: David Roe, Jean- | Reviewers: Jean-Pierre Flori,
Pierre Flori, Peter Bruin | Luca De Feo
Report Upstream: N/A | Work issues:
Branch: | Dependencies: #14958, #12142
Stopgaps: |
-------------------------------------+-------------------------------------
Changes (by pbruin):
* status: needs_work => needs_review
* work_issues:
store ref to PCL in finite field (or implement another way of weak
caching)
=>
Old description:
> Implements coercion within lattices of finite fields lying above the same
> prime when implemented with (pseudo-)Conway polynomials.
>
> {{{
> sage: k = GF(9, conway=True, prefix='z')
> sage: l = GF(27, conway=True, prefix='z')
> sage: x = k.gen() + l.gen(); x
> z6^5 + 2*z6^4 + 2*z6^3 + z6^2 + 2*z6 + 1
> sage: x.parent()
> Finite Field in z6 of size 3^6
> }}}
>
> This feature is implemented for fields outside the range of the Conway
> polynomial database by the implementation of a function for finding
> pseudo-Conway polynomials: polynomials that satisfy all of the algebraic
> constraints on Conway polynomials without the lexicographic constraint
> that imposes uniqueness.
>
> When using the `conway` and `prefix` parameters, one does not need to
> specify an explicit variable name; if no variable name is given, it is
> constructed from the `prefix` and the degree (as in the above code
> snippet).
>
> In the future, the functionality of this ticket will be incorporated into
> that for algebraic closures of finite fields.
>
> Apply:
> * [attachment:trac_8335-finite_field_coerce-5.11.b3-14888.patch]
> * [attachment:trac_8335-rebase_14958.patch]
New description:
Implements coercion within lattices of finite fields lying above the same
prime when implemented with Conway polynomials.
{{{
sage: k = GF(9, conway=True, prefix='z')
sage: l = GF(27, conway=True, prefix='z')
sage: x = k.gen() + l.gen(); x
z6^5 + 2*z6^4 + 2*z6^3 + z6^2 + 2*z6 + 1
sage: x.parent()
Finite Field in z6 of size 3^6
}}}
When using the `conway` and `prefix` parameters, one does not need to
specify an explicit variable name; if no variable name is given, it is
constructed from the `prefix` and the degree (as in the above code
snippet).
In the future, the functionality of this ticket will be incorporated into
that for algebraic closures of finite fields. It will then be possible to
construct compatible systems of finite fields outside the range of the
Conway polynomial database using the pseudo-Conway polynomials from
#14958: polynomials that satisfy all of the algebraic constraints on
Conway polynomials without the lexicographic constraint that imposes
uniqueness.
Apply:
* [attachment:trac_8335-finite_field_coerce-5.11.b3-14888.patch]
* [attachment:trac_8335-no_pseudo.patch]
--
--
Ticket URL: <http://trac.sagemath.org/ticket/8335#comment:112>
Sage <http://www.sagemath.org>
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