absolute_field() for towers of finite fields

[saraedum]

Towers of finite fields are hard to work with because they lose most of the nice functionality of simple extensions of prime finite fields and are also much slower.

Therefore it would be useful to have a method that turns a tower into a simple extension.

Note that there have been attempts to support relative extensions of finite fields in #21413 and that there is also sage.coding.relative_finite_field_extension, the state of which is unclear to me. There is also _isomorphic_ring on polynomial quotient rings which does something similar so this should probably be unified for finite fields.

Depends on #26105
Depends on #26161

CC: @sagetrac-swewers @roed314

Component: finite rings

Keywords: finite fields, polynomial quotient ring, padicBordeaux

Branch/Commit: u/saraedum/26103 @ 0f7c2c9

Issue created by migration from https://trac.sagemath.org/ticket/26103

[saraedum]

comment:1

See #25976 for a followup and MCLF/mclf#103 for some context.

[saraedum]

Description changed:

--- 
+++ 
@@ -3,3 +3,7 @@
 Therefore it would be useful to have a method that turns a tower into a simple extension. More generally there should be a function `simple_model(base=None)` on polynomial quotient rings that returns an isomorphic simple extension of `base` together with isomorphisms. `base` should default to the prime subfield.
 
 Initially this would only be implemented for trivial extensions and towers of finite fields.
+
+As an alternative, there could also be a `absolute_field()` method in the category of finite fields.
+
+Note that there have been attempts to support relative extensions of finite fields in #21413 and that there is also `sage.coding.relative_finite_field_extension`, the state of which is unclear to me.

[saraedum]

Branch: u/saraedum/26103

[saraedum]

New commits:

2fc28c2

Add absolute_field() for finite fields

[saraedum]

Commit: 2fc28c2

[saraedum]

Dependencies: #26105

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

14e6fdc

Comment on why this method should live in FiniteFields()

[sagetrac-git]

Changed commit from 2fc28c2 to 14e6fdc

[saraedum]

Description changed:

--- 
+++ 
@@ -1,9 +1,5 @@
 Towers of finite fields are hard to work with because they lose most of the nice functionality of simple extensions of prime finite fields and are also much slower.
 
-Therefore it would be useful to have a method that turns a tower into a simple extension. More generally there should be a function `simple_model(base=None)` on polynomial quotient rings that returns an isomorphic simple extension of `base` together with isomorphisms. `base` should default to the prime subfield.
-
-Initially this would only be implemented for trivial extensions and towers of finite fields.
-
-As an alternative, there could also be a `absolute_field()` method in the category of finite fields.
+Therefore it would be useful to have a method that turns a tower into a simple extension.
 
 Note that there have been attempts to support relative extensions of finite fields in #21413 and that there is also `sage.coding.relative_finite_field_extension`, the state of which is unclear to me.

[saraedum]

comment:7

Without proper support for morphisms with base_morphisms, the output is slightly ugly:

sage: k = GF(2)
sage: R.<a> = k[]
sage: l.<a> = k.extension(a^3 + a^2 + 1)
sage: l.absolute_field()
(Finite Field in z3 of size 2^3, Composite map:
   From: Finite Field in a of size 2^3
   To:   Finite Field in z3 of size 2^3
   Defn:   Ring morphism:
           From: Finite Field in a of size 2^3
           To:   Univariate Polynomial Ring in a over Finite Field of size 2 (using GF2X)
           Defn: a |--> a
         then
           Ring morphism:
           From: Univariate Polynomial Ring in a over Finite Field of size 2 (using GF2X)
           To:   Univariate Polynomial Ring in a over Finite Field in z3 of size 2^3
           Defn: Induced from base ring by
                 Composite map:
                   From: Finite Field of size 2
                   To:   Finite Field in z3 of size 2^3
                   Defn:   Identity endomorphism of Finite Field of size 2
                         then
                           Ring morphism:
                           From: Finite Field of size 2
                           To:   Finite Field in z3 of size 2^3
                           Defn: 1 |--> 1
         then
           Ring morphism:
           From: Univariate Polynomial Ring in a over Finite Field in z3 of size 2^3
           To:   Finite Field in z3 of size 2^3
           Defn: a |--> z3^2 + 1, Ring morphism:
   From: Finite Field in z3 of size 2^3
   To:   Finite Field in a of size 2^3
   Defn: z3 |--> a^2 + a)

Unfortunately, this does not work yet if we add another extension on top:

sage: R.<b> = l[]
sage: m.<b> = l.extension(b^2 + b + a)
sage: m.absolute_field()
AttributeError: no attribute absolute_field

The problem is that even though m is in the category of finite fields, it does not inherit the parent methods for some reason.

---

New commits:

cb8418a	Fix absolute_field() implementation
8685729	Some temporary hacks to make absolute_field() work