Changes between Version 3 and Version 4 of Ticket #14990


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Timestamp:
08/15/13 15:17:41 (8 years ago)
Author:
pbruin
Comment:

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  • Ticket #14990 – Description

    v3 v4  
    11The goal of this ticket is a basic implementation of algebraic closures of finite fields.  Most importantly, it provides the following:
    22
    3 - class `AlgebraicClosureFiniteField`
     3- class `AlgebraicClosureFiniteField_generic` (abstract base class)
    44  - 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
     5- class `AlgebraicClosureFiniteField_pseudo_conway` (implements the specific defining polynomials of the finite subfields and the relations between the generators of these subfields)
    56- class `AlgebraicClosureFiniteFieldElement`
    67  (mostly 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)
    7 - method `FiniteField.algebraic_closure()`
    8 
    9 (Alternative names: `FiniteFieldAlgebraicClosure`, `FiniteFieldAlgebraicClosureElement`, maybe with aliases `FFpbar`, `FFpbarElement`?)
     8- factory class `AlgebraicClosureFiniteField` (to get unique parents)
     9- method `FiniteField.algebraic_closure()` (invokes the factory class)
    1010
    1111An example using the new functionality is the following analogue of the example from #8335:
    … …  
    3737}}}
    3838
    39 (The above example shows that `AlgebraicClosureFiniteFieldElement` is printed in the same way as the underlying `FiniteFieldElement`.)
     39It is conceivable that there will be different coexisting implementations (deriving from `AlgebraicClosureFiniteField_generic`).  The current implementation uses Conway polynomials and the pseudo-Conway polynomials from #14958, as well as the functionality for finite field homomorphisms provided by #13214.
    4040
    41 It is conceivable that there will be different coexisting implementations (classes deriving from an abstract `AlgebraicClosureFiniteField`).  The current implementation uses Conway polynomials and the pseudo-Conway polynomials from #14958, as well as the functionality for finite field homomorphisms provided by #13214.
     41Apply: [attachment:trac_14990-algebraic_closure_finite_field.patch]