sage.arith: Use sage.rings.abc and move some module-level imports into methods

[mkoeppe]

(cherry-picked from #32432)

Depends on #32566

CC: @tscrim

Component: refactoring

Author: Matthias Koeppe

Branch/Commit: 5e5843a

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/32600

[mkoeppe]

Changed dependencies from #32566: to #32566

[mkoeppe]

Branch: u/mkoeppe/sage_arith__use_sage_rings_abc_and_move_some_module_level_imports_into_methods

[mkoeppe]

New commits:

cee081a	sage.rings.real_mpfr.RealField_class: Inherit through a new class sage.rings.abc.RealField
99f4f10	sage.rings.real_double.RealDoubleField_class: Inherit through a new class sage.rings.abc.RealDoubleField
29871d0	ComplexField_class, ComplexDoubleField: Inherit through new classes sage.rings.abc.*
e1bb944	src/sage/rings/polynomial/polynomial_element.pyx: Replace use of is_RealField by isinstance(sage.rings.abc.RealField) etc.
6e2c3c4	src/sage/rings/abc.pyx: Add class docstrings
2bd66e4	Merge #32566
4e2f0ab	src/sage/arith/misc.py: move pari/flint imports into methods
5145d43	src/sage/arith/misc.py: Use sage.rings.abc instead of importing element classes RealNumber, ComplexNumber
5e5843a	sage.arith.misc: Fixup

[mkoeppe]

Commit: 5e5843a

[mkoeppe]

Author: Matthias Koeppe

[tscrim]

comment:5

While it's not an issue here since it is not the main part of the algorithm, changes like this do incur a slight performance penalty:

-    if isinstance(z, (RealNumber, ComplexNumber)):
+    if isinstance(z.parent(), (RealField, ComplexField)):

This is just for future reference.

[tscrim]

Reviewer: Travis Scrimshaw

[mkoeppe]

comment:6

Thanks! I agree with your analysis.

[vbraun]

Changed branch from u/mkoeppe/sage_arith__use_sage_rings_abc_and_move_some_module_level_imports_into_methods to 5e5843a