Adding generic change_ring() method and some other touchups.

[tscrim]

We implement a generic change_ring() method.

That there is no such method was noticed in #36368.

We also do some other minor touchups: Use _repr_ instead of __repr__ and

📝 Checklist

• The title is concise, informative, and self-explanatory.
• The description explains in detail what this PR is about.
• I have linked a relevant issue or discussion.
• I have created tests covering the changes.
• I have updated the documentation accordingly.

⌛ Dependencies

[tscrim]

@roed314 This branch causes the following p-adic test to have a very long loop of changing ideals:

sage: W = Zp(3,10)
sage: S.<x> = W[]
sage: R.<pi> = W.extension(x^2 + 3)
sage: R.residue_field()(pi) == pi

I've spent the past 2 hours trying to figure it out, but I still don't understand what is going on. This is as small of an example as I was able to devise from the failing padic doctests. Even once the check figures out how it should be done, it creates a number of 0 elements in performing the check as opposed to before this branch. Any idea about what is going on here?

[tscrim]

Before:

sage: W = Zp(3,10)
sage: S.<x> = W[]
sage: R.<pi> = W.extension(x^2 + 3)
sage: R.residue_field()(pi) == pi
False

With the PR:

sage: W = Zp(3,10)
sage: S.<x> = W[]
sage: R.<pi> = W.extension(x^2 + 3)
sage: R.residue_field()(pi) == pi
True

With PR in a fresh session:

sage: W = Zp(3,10)
sage: S.<x> = W[]
sage: R.<pi> = W.extension(x^2 + 3)
sage: R(pi, absprec=20)
pi + O(pi^20)
sage: R(pi^3, absprec=20)
---------------------------------------------------------------------------
RecursionError                            Traceback (most recent call last)
[very long traceback]
RecursionError: maximum recursion depth exceeded while calling a Python object
sage: R(pi^3, absprec=20)
pi^3 + O(pi^20)
sage: R.residue_field()(pi) == pi
False

[roed314]

I'll try to take a look at this soon.

[tscrim]

Thanks. I have no idea whatsoever what is going wrong with this change. Any thoughts/pointers/help would be appreciated.