Changes between Version 3 and Version 4 of Ticket #14990
 Timestamp:
 08/15/13 15:17:41 (6 years ago)
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Ticket #14990 – Description
v3 v4 1 1 The goal of this ticket is a basic implementation of algebraic closures of finite fields. Most importantly, it provides the following: 2 2 3  class `AlgebraicClosureFiniteField `3  class `AlgebraicClosureFiniteField_generic` (abstract base class) 4 4  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) 5 6  class `AlgebraicClosureFiniteFieldElement` 6 7 (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) 10 10 11 11 An example using the new functionality is the following analogue of the example from #8335: … … 37 37 }}} 38 38 39 (The above example shows that `AlgebraicClosureFiniteFieldElement` is printed in the same way as the underlying `FiniteFieldElement`.) 39 It is conceivable that there will be different coexisting implementations (deriving from `AlgebraicClosureFiniteField_generic`). The current implementation uses Conway polynomials and the pseudoConway polynomials from #14958, as well as the functionality for finite field homomorphisms provided by #13214. 40 40 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 pseudoConway polynomials from #14958, as well as the functionality for finite field homomorphisms provided by #13214. 41 Apply: [attachment:trac_14990algebraic_closure_finite_field.patch]