sage.rings.factorint: Modularization fixes

[mkoeppe]

📚 Description

Spliiting sage.rings.factorint into separate modules, per library used.

Part of:

• Meta-ticket: Modularize sagelib into separate distributions (pip-installable packages) sagemath-... for Sage 10.x #29705

📝 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

[Drewshotwell]

This is looking good from my testing. The only thing I noticed is that in the Examples section it's still sage: from sage.rings.factorint import factor_using_flint when it should be sage: from sage.rings.factorint_flint import factor_using_flint

[mkoeppe]

Thanks for catching this; fixed now

[kwankyu]

Please add boilerplate head stuffs, especially copyright notice.

[mkoeppe]

Thanks, done.

[github-actions]

Documentation preview for this PR is ready! 🎉
Built with commit: e8051b8

[kwankyu]

Thanks. LGTM.

[mkoeppe]

Thank you!