#14990: Implement algebraic closures of finite fields
-------------------------+-------------------------------------------------
Reporter: pbruin | Owner:
Type: | Status: new
enhancement | Milestone: sage-5.12
Priority: major | Keywords: finite field algebraic
Component: algebra | closure
Merged in: | Authors:
Reviewers: | Report Upstream: N/A
Work issues: | Branch:
Dependencies: #14958 | Stopgaps:
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Algebraic closures of finite fields should be implemented. Most
importantly, we will need the following:
- class `AlgebraicClosureFiniteField`
- method `subfield(n)` returning a tuple consisting of the subfield of
order ''p^n^'' and a `RingHomomorphism_im_gens` giving the canonical
embedding into the algebraic closure
- class `AlgebraicClosureFiniteFieldElement`
(should mostly be a wrapper around `FiniteFieldElement`, so actually an
element of a finite subfield, but having the algebraic closure as its
parent and taking care of coercion into larger subfields)
- method `FiniteField.algebraic_closure()`
(Alternative names: `FiniteFieldAlgebraicClosure`,
`FiniteFieldAlgebraicClosureElement`, maybe with aliases `FFpbar`,
`FFpbarElement`?)
An example using the new functionality would be the following analogue of
the example from #8335:
{{{
sage: Fbar = GF(3).algebraic_closure('z')
sage: Fbar
Algebraic closure of Finite Field of size 3
sage: F2, e2 = Fbar.subfield(2)
sage: F3, e3 = Fbar.subfield(3)
sage: F2
Finite Field in z2 of size 3^2
}}}
To add, we first embed into `Fbar`:
{{{
sage: x2 = e2(F2.gen())
sage: x3 = e3(F3.gen())
sage: x = x2 + x3
sage: x
z6^5 + 2*z6^4 + 2*z6^3 + z6^2 + 2*z6 + 1
sage: x.parent() is Fbar
True
}}}
It would be nice to do this without explicitly invoking the embeddings; as
a shortcut, `Fbar` should have a method `gen(n)` returning the fixed
generator of the subfield of degree ''n'', but as an element of `Fbar`:
{{{
sage: x2 == Fbar.gen(2)
True
sage: x == Fbar.gen(2) + Fbar.gen(3)
True
}}}
(The above example assumes that an `AlgebraicClosureFiniteFieldElement` is
printed in the same way as the underlying `FiniteFieldElement`; we do not
necessarily have to do this.)
It is conceivable that there will be different coexisting implementations
(classes deriving from an abstract `AlgebraicClosureFiniteField`). The
first (and easiest) to implement should use Conway polynomials and the
pseudo-Conway polynomials from #14958, probably using some of the code
from #8335.
--
Ticket URL: <http://trac.sagemath.org/ticket/14990>
Sage <http://www.sagemath.org>
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