free_module method for finite fields, number fields and p-adics

[roed314]

In support of the ring extension classes in #21413, we want to generalize the functionality of the vector_space method to free modules over other rings. This ticket renames the method to free_module and implements it for p-adic extensions.

CC: @saraedum @xcaruso

Component: algebra

Keywords: padicBordeaux

Author: David Roe

Branch/Commit: 242fecc

Reviewer: Xavier Caruso

Issue created by migration from https://trac.sagemath.org/ticket/28481

[roed314]

Branch: u/roed/free_module

[roed314]

Author: David Roe

[roed314]

Changed keywords from none to padicBordeaux

[roed314]

Commit: 23947e9

[roed314]

New commits:

7d88f9a	Working on p-adic free module isomorphisms
f4f17d6	Fix precision problem in vector space maps by using new method _polynomial_list
23947e9	Working on vector_space and free_module

[tscrim]

comment:4

The free_module in padic_extension_generic.py does not have any doctests.

About this comment

# We could have used morphisms in the category
# FiniteDimensionalModulesWithBasis over Qp(p)
# But this would require making p-adic fields
# part of this category which has a lot of side
# effects (adding methods which will show up
# in tab completion for example).  For now we
# just stick with Map.

having extra methods via tab completion is a good thing, especially as they are all still applicable to the objects. That specific reason really is not a good argument for not doing that, but are there other side effects that cause problems?

-            - ``basis`` -- a basis of the field as a vector space
-              over the subfield.  If not given, one is chosen automatically.
+            - ``basis`` -- a basis of the field as a vector space
+              over the subfield; f not given, one is chosen automatically
 
-            - ``map`` -- whether to return maps from and to the vector space.
+            - ``map`` -- whether to return maps from and to the vector space

-        - ``basis`` -- a basis for the vector space (ignored)
+        - ``basis`` -- (ignored) a basis for the vector space
 
-        - ``map`` -- whether to return maps to and from the vector space (default True)
+        - ``map`` -- (default: ``True``) whether to return maps to and from the vector space

-        - ``base`` -- a subfield (default: ``None``), the returned vector
+        - ``base`` -- a subfield (default: ``None``); the returned vector
           space is over this subfield `R`, which defaults to the base field of this
-          function field.
+          function field
 
-        - ``basis`` -- a basis for this field over the base.
 
         - ``maps`` -- boolean (default ``True``), whether to return
-          `R`-linear maps to and from `V`.
+          `R`-linear maps to and from `V`

-        - ``basis`` -- a list of elements giving a basis over the subfield (optional)
+        - ``basis`` -- (optional) a list of elements giving a basis over the subfield
 
+        - ``map`` -- whether to return isomorphisms to and from the vector space (default ``True``)
+        - ``map`` -- (default: ``True``) whether to return isomorphisms to and from the vector space

Also Returns -> Return.

[sagetrac-git]

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

8db48a2

Reviewer suggestions

[sagetrac-git]

Changed commit from 23947e9 to 8db48a2