new issue after moving the zeta method

[fchapoton]

Steps To Reproduce

Some random failure in the CI can be reproduced by this :

sage: K=Integers(3659038768515778597)
sage: b=K(2734973578022464281)
sage: b.nth_root(3250669396930106370)
...
AttributeError: 'IntegerModRing_generic_with_category' object has no attribute '_element_of_factored_order'

Expected Behavior

either an answer or a ValueError

Actual Behavior

raise an AttributeError

Additional Information

No response

Environment

• OS:
• Sage Version:

Checklist

• I have searched the existing issues for a bug report that matches the one I want to file, without success.
• I have read the documentation and troubleshoot guide

[fchapoton]

probably we need to implement a zeta method in IntegerMod rings ?

[tscrim]

I confirmed this was working in 10.7.beta3 but then failed when pulling in #39532.

[tscrim]

The problem comes from the fact that _element_of_factored_order is only implemented for a particular concrete implementation of FiniteField, not for the general category. When calling the nth_root(), it is doing a primality check and refining the category to the field:

sage: K=Integers(3659038768515778597)
sage: K.category()
Join of Category of finite commutative rings and Category of subquotients of monoids and Category of quotients of semigroups and Category of finite enumerated sets
sage: b=K(2734973578022464281)
sage: b.nth_root(3250669396930106370)
...
AttributeError: 'IntegerModRing_generic_with_category' object has no attribute '_element_of_factored_order'
sage: K.category()
Join of Category of finite enumerated fields and Category of subquotients of monoids and Category of quotients of semigroups

which subsequently leads to the failure.

Thus, to fix the issue, we need to lift the _element_of_factored_order up to the category. However, considering that the method has some (mild) Cython type declarations, I think we should leave the Cythonized copy there as it could have a performance impact. Of course, if it can be demonstrated that it has no real impact, then we can remove it.

[fchapoton]

Thanks. I have made a proposal in #40170 by just copying the method in the category.