Hello,
I'm interested in moving the _nth_root_common method in
FiniteRingElement in src/sage/rings/finite_rings/element_base.pyx to the
FiniteFields category's ElementMethods. This would enable Sage to take
nth roots in a finite field constructed by quotienting a ring of
polynomials over a finite field by an irreducible polynomial, and also
get a bit closer to eventually having all finite field stuff in the same
part of the codebase rather than split between
src/sage/rings/finite_rings and src/sage/categories.
I think this should be a pretty straightforward change to make, but I
have some questions since _nth_root_common is written in Cython and
there seems to be very little use of Cython currently in
src/sage/categories.
1. Are there any technical limitations or restrictions on using Cython
methods in the category framework? In
https://doc.sagemath.org/html/en/reference/structure/sage/structure/dynamic_class.html
there's a mention of possible issues with Cython and dynamic
classes, but it's not clear to me what the implications of that are.
2. At least of the categories I've looked at (mainly FiniteFields and
related categories), it doesn't look like we've implemented many
computationally difficult functions in ElementMethods. Is there a
technical reason for this or it it just that nobody's gotten around
to doing this work yet?
3. Is there any benefit to having _nth_root_common implemented in
src/sage/rings/finite_rings/element_base.pyx rather than making a
new file such as src/sage/categories/finite_field_elements.pyx and
moving it there?
4. Any other advice/warnings before I look into this further?
--
Vincent Macri (he/him)
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