absolute_field() for towers of finite fields

[Julian Rüth]

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.

[Julian Rüth]

See #25976 for a followup and ​https://github.com/MCLF/mclf/issues/103 for some context.

[Julian Rüth]

New commits:

​2fc28c2

Add absolute_field() for finite fields

[git]

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

​14e6fdc

Comment on why this method should live in FiniteFields()

[Julian Rüth]

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

[git]

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

​8db7302

Merge remote-tracking branch 'trac/develop' into 26103

[Julian Rüth]

It only gets worse for larger towers:

sage: R.<b> = l[]
sage: m.<b> = l.extension(b^2 + b + a)
sage: m.is_field() # refines category
sage: m.absolute_field()
(Finite Field in z6 of size 2^6, Composite map:
   From: Univariate Quotient Polynomial Ring in b over Finite Field in a of size 2^3 with modulus b^2 + b + a
   To:   Finite Field in z6 of size 2^6
   Defn:   Ring morphism:
           From: Univariate Quotient Polynomial Ring in b over Finite Field in a of size 2^3 with modulus b^2 + b + a
           To:   Univariate Polynomial Ring in b over Finite Field in a of size 2^3
           Defn: b |--> b
         then
           Ring morphism:
           From: Univariate Polynomial Ring in b over Finite Field in a of size 2^3
           To:   Univariate Polynomial Ring in b over Finite Field in z6 of size 2^6
           Defn: Induced from base ring by
                 Composite map:
                   From: Finite Field in a of size 2^3
                   To:   Finite Field in z6 of size 2^6
                   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 + 1
                         then
                           Ring morphism:
                           From: Finite Field in z3 of size 2^3
                           To:   Finite Field in z6 of size 2^6
                           Defn: z3 |--> z6^5 + z6^4 + z6^2 + 1
         then
           Ring morphism:
           From: Univariate Polynomial Ring in b over Finite Field in z6 of size 2^6
           To:   Finite Field in z6 of size 2^6
           Defn: b |--> z6^4 + z6^3, Ring morphism:
   From: Finite Field in z6 of size 2^6
   To:   Univariate Quotient Polynomial Ring in b over Finite Field in a of size 2^3 with modulus b^2 + b + a
   Defn: z6 |--> (a + 1)*b + a^2 + a)

[git]

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

​c68e16d

Hack around problems in morphisms of quotients and finite fields

[git]

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

​12fd855	Refine polynomial quotient ring to the finite subcategory on initialization
​0f7c2c9	Merge branch '26161' into 26103

[Julian Rüth]

​https://gitlab.com/sagemath/sage/-/merge_requests/33 implements this in a more general way it seems.

So what's been done here is probably irrelevant.