Add absolute_degree method to finite fields

[r-mb]

This PR adds an absolute_degree method to various finite fields class. This is done in order to fix is_prime_field. The reason is explained in my bug report #40426.

In the same time, I also updated documentation is various parts of the code.

Fixes #40426.

📝 Checklist

• The title is concise and informative.
• 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 and checked the documentation preview.

[user202729]

To be fair, number fields have .degree(), .absolute_degree(), .relative_degree() too, with absolute degree being the degree over ℚ (the prime field in characteristic zero case).

Maybe we could have something similar here.

[r-mb]

I agree that this would be the best solution. The thing is that, to have something similar to number fields, I believe we would either need to implement extensions of finite fields, which would require much work, or to do a breaking change.

Currently, SageMath uses the extension method of the CommutativeRing class, which in turn creates a PolynomialQuotientRing_field, which is a subclass of PolynomialQuotientRing_generic which defines degree.

If we rename degree there to relative_degree we would need to first do deprecation warnings isn't it? And then we could add an absolute_degree method in PolynomialQuotientRing_field and FiniteField.

[user202729]

Not necessarily, we could just add absolute_degree which always mean degree over the prime field. Then modify is_prime_field to rely on absolute_degree.

[r-mb]

Sorry for the delay, I'm on holiday and will be slow to answer in the next weeks (but I will always reply eventually).

@user202729 I liked your idea to add absolute_degree very much, so I did so! This made me realise that the bug I fix in this PR also existed for extensions of QQ, so I also fixed that and added relevant doctests.

[r-mb]

Failing doctests, please wait until I can fix this.

[r-mb]

Ready for review!

[github-actions]

Documentation preview for this PR (built with commit b69ee84; changes) is ready! 🎉
This preview will update shortly after each push to this PR.

[user202729]

I... guess I forgot this. Looks reasonable. I don't like the isinstance() and hasattr but there might not be a better way (duck typing).

(self in Fields() and self.absolute_degree()==1 might work, but what about something like Laurent polynomial ring which is a field of infinite degree over ℚ? Or say $ℚ_p$?)

[r-mb]

Thank you!
Indeed having to do these checks is not the best, but by doing the other test we would need to implement absolute_degree everywhere, and make sure that every future new class of fields also do...